Discrete Differential Geometry: Integrable Structure (Graduate Studies in Mathematics 98)

)

(4.46)

sin ~(T2 + - <j)+ - ti4> + <j>) =

tan(c*2 / 2 ) tan( 7 / 2 ) sin ^ ( t 2 + + r 2 (j) + 0 ).

Here the Backlund parameter 7 comes from cos 7 = (n + , n ) .

4.2.6. D iscrete sine-Gordon equation. In this section, we study the angles between adjacent edges of a discrete K-surface / : 1? —> R3. It will be assumed that the surface / is immersed, that is, in each tangent plane the four edges incident to the vertex / are cyclically ordered as in Figure 4.6, and the angle between any pair of consecutive edges is in (0, tx). Each

4.2. Discrete K-nets

143

elementary quadrilateral (/, / i , / 12, / 2) has, due to the Chebyshev property, two pairs of equal angles, (4.47)

V = ^ ( / / i> // 2 ) = A / 1 2 / 2 , / 1 2 / 1 )

and (4.48) It turns out to be possible to express these angles through the function : Z 2 —> R introduced in the previous section and satisfying the Hirota equation (4.35).

/2

fl2

Si

Figure 4.6. Four quadrilaterals of a discrete K-net adjacent to the vertex / , and their inner angles.

Proposition 4.17. (Edges of a discrete K-surface) Up to the common rotation _1(- ■•)$, the edges of a discrete K-surface lying in the tangent plane at the vertex f are given by:

(4.49)

f i - f

=

-isin c*! ( y (01_ 0)/2

(4.50)

h ~ f

=

- * s in a 2 I ^ - i { M ")/2

0

0

0

ei(2+)/'Z'' 0 i

)/2N

4. Special Classes of Discrete Surfaces

144

(In the first two formulas a \ and a 2 are the labels assigned to the edges ( u, u + ei) and ( u, u + e2), respectively, while in the last two formulas they are the labels assigned to the edges (u — e \ , u) and (u — e 2 ,u), respectively.)

Proof. Since the derivation is the same for all four formulas, we only give details for the first two of them. From the discrete Lelieuvre representation (4.32) and expression (4.12) for the Gauss map n through the frame <E> we derive: f i - f

=

ni x n = ^ [^^f1e 3^ i , ^► _ 1e 3^>] = ^ $ _ 1 [{7 f le 3t / i ,e 3]$,

h ~ f

=

n x n 2 = ^ [$ - 1e 3$ , $ J :l e 3^ 2\ = ^ $ _ 1[e3, C/2“ le 3^ 2]^-

Substituting e 3 = —ias and expressions (4.33), (4.34) for the transition matrices, we obtain the formulas

[Ui

e3t/i,e3] = _1

[e3, U2 e3U2] =

-ts in a !

Q (

-isina2 I

J,

eill2\

0

Q J,

which coincide with (4.49), (4.50).

□

Corollary 4.18. (A ngles betw een th e edges o f a discrete K-surface) In the notation of Figure 4.6, (4.53)

V^-1,-2 = g (0-1 + 0-2)>

(4.54)

(p*_l = T r - ^ ( 0 2 + 0 - 1),


Proof. Again, all four formulas are verified analogously; therefore we give the details only for the first one. Upon using formula (4.10) for the scalar product of the su( 2 )-representatives of vectors in M3, we find: COS (f

(fi ~ / , h ~ f )

=

\\fi - f 1 -

2

tr

_ -tr (/i - f ) ( f 2 - /)

|| * 11/2 - /||

( e ~i(i + M / 2

^ 0

V

2 sin a i sin a 2

0

\

= cos-

ci(*i+fc)/2; -

01 + 02

2

'

This proves the first equation in (4.53). In a similar fashion, one shows that

0 - 1 + 0-2

COS(^_i _2 = C O S ---- ----- ,

02 + 0 -1

COS ( p _ ± = — c o s --- ---- ,

z

This completes the proof.

*

01 + 0 -2

cos ( p _ 2 — — c o s ---- ---- .

z

□

4.3. Discrete isothermic nets

145

As a consequence, we can derive a difference equation which governs the angles (p between the asymptotic lines of a discrete K-surface:

Theorem 4.19. (D iscrete sine-G ordon equation) The angles

..( w - i- v - a + v - ! ,- ,) = 1 - K-ie ^ . 1 - K - 2e **>-* ' 1 —K - \ e 1^ - 1 1 —K-2elv~2 '

e

Here notation is as in Figure 4.6, and k - i , k - 2 (ire the values of

(4.56)

k

— tan

tan ^

for the quadrilaterals (/__i, / , / 2, / —1,2) and ( / —2, f i - 2 , / 1, / ) , respectively.

Proof. We rewrite Hirota equation (4.35) for the quadrilateral (/, / 1, / 12, f 2) as I _ e-i(0i2+0-0i-02)/2 _ ^^ei(0i+02)/2 _ e-i(<^i2+0)/2^ where k is given in (4.56) and can be considered as assigned to this quadri­ lateral. According to (4.53), (4.54), we have: (4.57)

(f = i ( 0 i + 0 2 ),


Thus, the Hirota equation leads to a relation between the angles

1 - Kei(p 1 — hie With this relation, the discrete sine-Gordon equation (4.55) follows directly from planarity of the vertex star: e itp+iip*

(4.58)

k

— elv

(p +
^

4.3 . D isc r e te iso th erm ic n e ts 4.3.1. N otion of a discrete isotherm ic net. D efinition 4.20. (D iscrete isotherm ic net) A discrete isothermic net is a circular Koenigs net, i.e., a circular net f : Zm —> R N admitting a dual net /* : Zm —» R N in the sense of Definition 2.22. We can use the characterizations of Koenigs net derived in Section 2.3 in order to find characterizations of discrete isothermic nets. To this end, let / : Zm —> R n be a circular net. Then its lift f — f + eo + \f\2^oo into the light cone L ^4*1,1 satisfies the same equation (2.1) as the net / itself. In particular, / is a Koenigs net in R N if and only if / is a Koenigs net in R N+ l ^.

4. Special Classes of Discrete Surfaces

146

Projectively invariant characterizations of Koenigs nets / in R ^-1"1,1 im­ mediately translate into Mobius-geometric characterizations of isothermic nets / in R N . In this context, conditions such as “points / lie in a ddimensional space” should be understood as “vectors / span a (d + 1 )dimensional linear subspace”, and this is translated as “points / belong to a (d — l)-dimensional sphere”; see Section 9.3.2. Translating in this fashion Theorem 2.27, applied to a two-dimensional Koenigs net / in Myv+ia, into the language of Mobius geometry in R N , we come to the following statement.

Theorem 4.21. (5-point spheres for discrete isotherm ic surfaces) 1) A two-dimensional circular' net f : Z2 —> R N that does not lie in a 2-sphere is discrete isothermic if and only if for every u G Z2 the five points f and / ± i ,±2 He on a two-sphere not containing the four points f ± [, f ± 2 . 2) A two-dimensional circular net f : Z2 —> S 2 C R N in a 2-sphere is discrete isothermic if and only if for every u G Z2 the three circles through the point / , P) = circle(/, / , 2, / - 1,2 ),

C<down) = circle(/, /,,_ 2, / _ lf_2),

C (‘) = c ir c le ( /,/i,/_ 1), have one additional point in common, or. equivalently , the three circles through f ,

C = circle(/, /_ i,2, /_ i,_ 2),

= circl e (/, / i ,2, / 1. - 2 ),

C (2) = circle ( / , / 2, / _ 2), have one additional point in common.

F igure 4.7. Four circles of a generic discrete isothermic surface, with a 5-point sphere.

4.3. Discrete isothermic nets

147

F igure 4.8. Four circles of a planar (or spherical) discrete isothermic net.

Cases 1), 2) of Theorem 4.21 are illustrated in Figures 4.7, 4.8, respec­ tively. Similarly, we can translate Theorem 2.29, applied to a multidimensional Koenigs net / in R ^+1,1, into the language of Mobius-geometric properties of the net / in R^. We get the following statement.

T heorem 4.22. (M ultidim ensional discrete isotherm ic nets) A cir­ cular net f : Zm —►R^ is discrete isothermic if and only if for every elementary hexahedron of the net its four white vertices are concircular, and its four black vertices are concircular (each of these conditions implies the other).

Upon a stereographic projection, an elementary hexahedron of a discrete isothermic net leads to the Clifford configuration; see Clifford’s first theorem in Exercise 3.11. If the pole of the stereographic projection is chosen at / , then one arrives at the configuration described in Exercise 4.12.

4.3.2. Cross-ratio characterization of discrete isotherm ic nets. An­ other characterization of discrete isothermic surfaces can be given in terms of the cross-ratios. Recall that for any four concircular points a , b ,c , d £ R N their (real-valued) cross-ratio is defined by equation (1.77), with Clifford multiplication in the Clifford algebra G£(RN). The Clifford product of x , y G R^ satisfies xy + yx = —2( x, y) , and the inverse element of x G R^ in the Clifford algebra is given by x ~ l = —x/ \ x\ 2. Alternatively, one can identify the plane of the quadrilateral (a, 6, c, d) with the complex plane C,

148

4. Special Classes of Discrete Surfaces

and then multiplication in (1.77) can be interpreted as the complex multi­ plication. An important property of the cross-ratio is its invariance under Mobius transformations. For discrete isothermic surfaces Theorem 4.21 yields the following char­ acterization.

Theorem 4.23. (Cross-ratios of four adjacent quadrilaterals)) A two-dimensional circular net f : 1? —> R N is a discrete isothermic surface if and only if the cross-ratios q — q ( f , / i , / 12, f 2) of its elementary quadri­ laterals satisfy the following condition:

(4.59)

q • 9 - 1 -2 = q - 1 • 9 - 2 -

Here, as usual, the negative indices —i denote the backward shifts r ~ l , so that, e.g., q- 1 = q ( f - i J J 2 , f - i i2); see Figure 4.9.

Figure 4.9. Four adjacent quadrilaterals of a discrete isothermic sur­ face: the cross-ratios satisfy q q~ 1 , - 2 = q~i • q- 2 -

Proof. Perform a Mobius transformation sending / to 00 . Under such a transformation, the four adjacent circles through / turn into four straight lines ( f ± i f ± 2), containing the corresponding points /± i,± 2 - The cross-ratios turn into ratios of directed lengths, e.g., 9 (/? /ij/i,2 ,/2 ) = - »;y

^

If the affine space through the points f ± 1 , f ± 2 is three-dimensional, then, according to part 1) of Theorem 4.21, the four points / ± i ,±2 lie in a plane

4.3. Discrete isothermic nets

149

(a two-sphere through / = oo). Generalized Menelaus’ theorem (Theorem 9.12) provides the following necessary and sufficient condition for this, ,4 6 m 1 ‘

KhJl,2) j

l(fl,2,h)

l ( f —2i

2)

l ( f h -2, f - 2) ' K f -1,- 2 J - l )

* ( / - ! , / - 1 , 2) / ( / - l , 2 , / 2)

'

This is equivalent to (4.59) with f = 0 0 . If, on the contrary, the four points f ± 1, f ± 2 are coplanar, then, according to part 2) of Theorem 4.21, both lines (/-i,2 /i,2 ) and ( / _ i j_2/i,-2) meet the line ( f - i f i ) at the same point p ^ . Thus, we are in the situation of Figure 4.10, described by the Desargues theorem. Here, we apply the Menelaus the­ orem twice, to the triangle A ( /_ i, / 2, / 1) intersected by the line (/-i,2/i,2)> and to the triangle A ( / _ i , / _ 2 , / 1) intersected by the line ( / _ i ?_ 2/ i , - 2)‘ *(/2,/l2)

l ( f - l j - 1,2)

1(PW J - 1)

Kf-2,fl,-2)

l(fl2,fl)

K / - l , 2 , / 2)

/ ( / l , ^ 1))

l ( f l , —2i f l )

l { f —l , —2i / —2)

This yields formula (4.60), again.

□

Figure 4.10. Desargues theorem.

For multidimensional discrete isothermic nets Theorem 4.22 yields a sim­ ilar characterization.

Theorem 4.24. (Cross-ratios of three adjacent quadrilaterals) A circular net f : Zm —> is discrete isotherm ic if and only if the cross­ ratios of its elem entary quadrilaterals satisfy the following condition:

(4.61)


fo r any triple o f different indices i , j , k .

4. Special Classes of Discrete Surfaces

150

Proof. Again, perform a Mobius transformation sending / to oo. Under such a transformation, the three adjacent circles through / turn into three straight lines ( fi fj ), (fj fk) and (fkfi), containing the (white) points fij, f jk and fki , respectively. Concircularity of these white points with / means simply that they are collinear. The necessary and sufficient condition for this is given by the Menelaus theorem: (A

K fji fij) J

Kfki fjk)

K f i j J i ) ' K f j k J j )

’

K fii fki)

_

K fte J k )

~

’

Since the Mobius-invariant meaning of the ratios of directed lengths is given by the corresponding cross-ratios, Q i f J i J i j J j ) = ~ i j j 3 ' f 3y

equation (4.62) is equivalent to (4.61).

□

The conclusions of Theorems 4.23, 4.24 can be summarized as follows.

Theorem 4.25. (Factorized cross-ratios) A circular net f : Zm —►R N is discrete isothermic if and only if the cross-ratios of its elementary quadri­ laterals satisfy

(4.63)

q(fjijijjj) = — , a j

where cti (i = 1 , . . . , m) constitute a real-valued labelling of the edges of Zm (depend on Ui only).

Thus, both edges ( u, u + e*) and (u + ej, u + e* + ej) of an elementary square 6^ of Zm carry the label a* = ai (u) = ai ( u + ej), and, similarly, the other edges (u, u -j- ej) and (u + e*, u + e* + ej) both carry the label OLj — OLj(u) — otj(u + ei)\ see Figure 4.11. In the next subsection we will give a more concrete way of determining a* for a given discrete isothermic net.

Figure 4.11. Labelling of the edges of a discrete isothermic net.

4.3. Discrete isothermic nets

151

It should be mentioned that if a labelling a? of edges is given, then equa­ tion (4.63) defines a 2D system: the values of / , / ?;, fj uniquely determine f i j . The very existence of multidimensional discrete isothermic nets relies on the following fundamental property of the cross-ratio equation.

Theorem 4.26. (3D consistency of discrete isotherm ic nets) The 2D cross-ratio equation (4.63) is SD consistent for amj labelling a t of the edges. We will provide two completely different proofs of this theorem. A con­ ceptual proof related to discrete Moutard nets in quadrics, will be presented in Section 4.3.5. A direct computational proof, with a simultaneous deriva­ tion of the zero curvature representation, will be given in Section 4.3.7. As­ suming for a while the validity of Theorem 4.26, we can now formulate the following initial data which determine a discrete isothermic net completely: ( if ) the values of / in the coordinate axes (i — 1 , . . . , rn), i.e., rn discrete curves f \ ^ t with a common intersection point /(()); (L^) rn functions a-i : S* —> R on the edges of the coordinate axes $ i for i = 1 , . . . , rn. Notice that the data (L^) give rise to an edge labelling of Zm, by ex­ tending them to all other edges according to 77a j = a j.

4.3.3. D arboux transform ation of discrete isotherm ic nets. As usual, the multidimensional consistency of discrete isothermic nets leads to their transformations. D efinition 4.27. (D iscrete Darboux transform ation) A pair of dis­ crete isothermic nets /, / + : Zm —» with likewise factorized cross-ratios, such that (4.63) holds and (4.64)

aj

.

is related by a Darboux transformation if the cross-ratios of the “vertical” elementary quadrilaterals are also factorized:

(4.65)

q(f, fi,/+ ,/+ ) = ^ ,

i = 1....... m,

with some c E R. The net / + is called a Darboux transform of the net f with the parameter c.

Clearly, the following data determine a Darboux transform / + of a given discrete isothermic net / uniquely: ( D f ) a point / + (0); (D^) a real number c, the parameter of the transformation.

4. Special Classes of Discrete Surfaces

152

The definition of Darboux transformations means that if we set F ( u , 0) = /(u ), F(u, 1) = f + (u), then F : Zm x {0,1} —> R N is an M-dimensional isothermic net, where M = m + 1. In particular, F is a circular net, so that the Darboux transformation is a particular case of the Ribaucour transfor­ mation. The parameter c plays the role of the function attached to all the edges parallel to the M-th lattice direction. The multidimensional con­ sistency of the cross-ratio equation immediately translates into the following fundamental statement.

Theorem 4.28. (Perm utability of discrete D arboux transform a­ tions) Let f be a discrete isothermic net, and let f ^ be two of its Darboux transforms, with parameters c\, c2, respectively. Then there exists a unique discrete isothermic net f (12) which is simultaneously a Darboux transform of f ^ with parameter c2 and a Darboux transform of f ^ with parameter c\. The net f ( 12^ is uniquely determined by the condition that the corresponding points of the four discrete isothermic nets are concircular and have a constant cross-ratio

4.3.4. M etric of a discrete isotherm ic net. Now we turn to the char­ acterization of discrete Koenigs nets given in Theorem 2.30. Applied to circular nets, it says that such a net / is Koenigs if and only if there exists a function s : Zm —> M* such that for any circular quadrilateral (/, f i , f i j , f j ) with M the intersection point of diagonals, l ( M, f i j ) _ Sij _ Sj (4.66) l(MJ) ~ s ’ ~ Si (Note that the notation s comes to replace u which we reserve for general Koenigs nets.) The function s for circular nets has an additional property, which justifies calling it a metric coefficient.

Theorem 4.29. (M etric coefficient of discrete isotherm ic nets) For a discrete isothermic net f , relations (4.66) define a function s : Zm —►R* uniquely, up to a black-white rescaling (s Xs at black points, s i—> jis at white points), which can be fixed by prescribing s arbitrarily at one black and at one white point. There exists a labelling a of edges of Zm such that

(4.67)

|/j - / | 2 = aiSSi

(z = 1 , . . . , m).

A black-white rescaling of the function s results in the rescaling a i—►(\fi)~ 1a

of the labelling a.

Proof. For a circular quadrilateral (/, f i , f i j , f j ) with M the intersection point of diagonals, one has two pairs of similar triangles, A (/, f u M ) ~ A ( f j , f i j , M ) ,

A ( /, fj , M ) ~ A ( f u f {j, M);

4.3. Discrete isothermic nets

153

cf. Figure 4.12. Hence, _

_____ _ 1fij fj I \ Mf \ ~ |/i - / |

\ Mf j \

IM f j | _ |f . f j I2 \Mf\ I/i - /I2

\ Mfi j \ \ Mf \

\Mfi\

_

\ Mf \

\fj

- f\

It follows that (4.69)

\ Mf \

This can be written as (4.70) l ( M, f j ) l ( M, fj ) \f fj I2 l ( M, f ) l ( M, f ) \fi-f\<

l ( MJ i j ) l (M,

f)

\ Mfi \ _ \ f j —f \Mfj\

\fj -

/|5

l ( M, f i ) _ \ f n - f i l (M,

fj)

\ f j- f \ i

Indeed, contemplating Figure 4.12, it is not difficult to realize that the frac­ tions on the left-hand side of each of the two equations in (4.70) are ei­ ther both negative (for an embedded quadrilateral), or both positive (for a non-embedded quadrilateral), so that the replacement of the quotients of lengths in (4.69) by the quotients of directed lengths in (4.70) is legitimate. Substitute the defining relations (4.66) of the function s into (4.70):

F ig u r e 4 .1 2 . Embedded (left) and nonembedded (right) circular quadrilaterals.

(A 71

s j s ij

’

SSi

_

I

fij

~

f j \2

SjSjj

If i - / I 2 ’

SSj

_

1f j j

~

f j \2

If j - / I 2 ■

But this is equivalent to the claim that the functions (4.72)

OLi = SSi

possess the labelling property, Tjai — a*.

□

4. Special Classes of Discrete Surfaces

154

The notation a* for edge labellings in Theorems 4.25 and 4.29 is the same for a reason.

Theorem 4.30. (Origin of the edge labelling for factorized cross­ ratios) If the edge labelling a i fo r a discrete isotherm ic net f : Zm —> R n is introduced according to equation (4.67), then the cross-ratios of its elem entary quadrilaterals are factorized as in equation (4.63). Proof. For a circular quadrilateral (/, n(f

f

f

f \

fi, f i j

....- I-ft ~

ft

,

fj)

one has:

' I-fa ~ ^

where e = —1 for an embedded quadrilateral and e = 1 for a nonembedded one. Thus, a( f

f

f

=

1/i ~ /I

^

' \fij

fir

Upon using equations (4.67) and (4.68), the previous formula can be rewrit­ ten as /£ £ r \ _ a *s i 1M f j 1 _ a i s i mK M ) f j ) Qyjljil ji jl jj ) J

| ii /f p |

ajSj

J

\M fi\

ajSj

7/71/T i* \ ,

l ( M J i )

and finally, due to (4.66), we arrive at aiSi

Sj

ai

aj Sj

Si

aj

which proves the theorem.

□

Remark. Theorem 4.29 as it stands cannot be reversed: the existence of a function s satisfying (4.67) does not yield the Koenigs property. Indeed, from (4.67) and (4.70) one finds: / a 73 n { ' ’

KM, f a ) l ( M, f )

l ( M, f j ) = »j»ij l ( M, f i ) ssi ’

l ( MJ j j ) l(MJ)

l ( M, f j ) = SjSjj l(MJj) SSj ’

which are equivalent to (A 74 ^ 1 '

fi j ) _

l(MJ)

~

, Sjj_ 8

l ( M, fj)

’

l(MJj) ~

s3

(with the same sign ± in both equations). The latter equations are somewhat weaker than (4.66), which is necessary and sufficient for the net / to be Koenigs. However, assuming some additional information about / , it is possible to force the plus signs in the latter formulas. For instance, if it is known that all elementary quadrilaterals of a two-dimensional circular net / are embedded, then property (4.67) is sufficient to assure that / is Koenigs. Indeed, in this case a 2 / a \ < 0, so that (4.67) yields s 2 / s \ < 0 and 512/5 < 0 , and then the plus sign has to be chosen in (4.74).

4.3. Discrete isothermic nets

155

Remark. For a Darboux pair of discrete isothermic nets, formula (4.67) for the (m + l)-st direction reads: (4.75)

| / + - / | 2 = css+ ,

and literally coincides with the corresponding formula (1.76) for the smooth case.

4.3.5. M outard representatives of discrete isotherm ic nets. The metric coefficient of a discrete isothermic net / can be used to produce its Moutard representative by a suitable rescaling of its lift / into the light cone of R n + 1 '1. This leads to a new characterization of discrete isothermic nets, which is manifestly Mobius invariant, since it is given entirely within the formalism of the projective model of Mobius geometry. The following statement is a discrete analog of Theorem 1.32. Theorem 4.31. (D iscrete isotherm ic net = T -net in light cone) If f : Zm —> WN is a discrete isotherm ic n et, then its lift s = s ~ l f : Zm —► L^4-1,1 to the light cone of R N+1,1 satisfies the discrete M outard equation (4.76)

nTjS - s = ai j (rj s -

tis ).

C onversely, given a discrete T-net s : Zm —> h N+l t l in the light cone, let the fu nctions s : Zm —>R and f : Zm —>R^ be defined by

(4.77)

s = s _1( / + e0 + | / | 2eoo)

(so that s ~ l is the e^-component, and s-1 / is the RN -part of s in the basis e i , . . . , eyv, eo, eooj- Then f is a discrete isotherm ic net.

Proof. This follows from Theorem 2.32 and the fact that for a Koenigs net / in R n the net / = / + e0 + | / | 2eoo is also a Koenigs net in the light cone Thus, we found an interpretation of discrete isothermic nets as an in­ stance of T-nets in a quadric, governed by (4.76) with _

(4.78)

=

( I ,

1J

TjS -

TjS)

i n s , TjS)

The edge labelling of a discrete isothermic net / (which provides the factor­ ization (4.63) of its cross-ratios) is already encoded in its lift s to the light cone. Indeed, OLi —

SSi

— 2(5, TiS],

and Theorem 4.5 assures that these quantities depend on Ui only. Theorem 4.3 says that T-nets in a quadric are 3D consistent. This proves the 3D consistency of discrete isothermic nets (Theorem 4.26). In particular,

4. Special Classes of Discrete Surfaces

156

Darboux transformations are nothing but Moutard transformations in the light cone. They are governed by the discrete Moutard equations (4.79)

Ti§+ - s = bi(s

- ns ) ,

bi =

(ns, s+)

and are specified by prescribing the value s+ (0) at one point. The parameter c of a Darboux transformation is encoded in the quantity c _ _ 2(m + > _ ! £ z 4 ! , SS^ which is independent of u G Zm, so that c = —2(s(0), s+ (0)).

4.3.6. Christoffel duality for discrete isotherm ic nets. Specializing the notion of Christoffel duality from general Koenigs nets to circular ones, the first essential observation is: the dual net for a discrete isothermic net is discrete isothermic, as well. Indeed, any quadrilateral with sides parallel to the corresponding sides of a circular quadrilateral is, obviously, also cir­ cular. A more detailed description of duality for discrete isothermic nets is contained in the following theorem. Theorem 4.32. (Christoffel dual of a discrete isotherm ic net) Let f : Zm —>R n be a discrete isothermic net, with factorized cross-ratios (4-80)

= — aJ

and with metric coefficient s : Zm —> R*. one-form Sf* defined by

(4.81)

Sif* = a i j ^ j -2 = — , \ViJ\

Then the R N -valued discrete

i = 1 , . . . , m,

is exact. Its integration defines (up to a translation) a net /* : Z 2 —►WN , called Christoffel dual of the net f . The net /* is discrete isothermic, with cross-ratios

(4 .82 )

= a3

and with metric coefficient s* = s _1 : Zm —> R*. Conversely, if for a given Q-net f : Zm —> R^ there exists an edge labelling a* such that the discrete one-form

(4-83)

sif* = a l - ^

is exact, then f is a discrete isothermic net, with cross-ratios as in (4.80).

4.3. Discrete isothermic nets

157

Proof. The first part of the theorem is a consequence of the general con­ struction of dual Koenigs nets. Equation (4.82) follows directly from (4.80), (4.81). To prove the converse part, identify the plane of the quadrilat­ eral (/, fi, fij, f j ) with C. Then the exactness condition for an elementary quadrilateral is equivalent to (the complex conjugate of) OLi

OLi

_

fi ~~ f

fij ~ f j

OLj

OLj

fj ~ f

fij ~ fi

Upon clearing denominators the latter equation turns into the cross-ratio equation (4.80) (in the generic situation, when fij ~ fi —fj + f ^ 0). Thus, the exactness of the form (4.83) actually characterizes discrete isothermic nets. □

Corollary 4.33. The noncorresponding diagonals of any elementary quadri­ lateral of a discrete isothermic net f and of its Christoffel dual are related by

(4.84)

/*-/* =

«,•)j |r £ y j 2 >

(m -

- /* = (<* -

•

Proof. We start with putting (4.63) into several equivalent forms; these computations hold not only in the Clifford algebra C£(RN) but also in an arbitrary associative algebra with unit A. Written as (4.85)

atiifij -

-

f ) ~ l =

OLj ( f i j -

fj)(fj

-

/ )

“

\

this equation displays the symmetry with respect to the diagonal flips of an elementary quadrilateral, expressed as fi f j and / respectively (both have to be accompanied by the change ol{ ctj). Writing (4.85) as aiifij ~

f)(fi

~

/r

1

-

OLi

=

OLjifij

-

f)(fj

-

f ) - 1 -

aij,

and dividing from the left by fij — / , we arrive at the so-called three-leg form of the cross-ratio equation: (4.86)

(a,

-

OLj){fij -

f ) ~ l

=

atiifi -

/ )

- 1

-

OLj(fj -

f ) ~ l .

Observe that, according to (4.81), the right-hand side of (4.86) is equal to ~( f i ~ /* ) + (fj ~ /* ) — / / —fi - This proves the first equation in (4.84), if one takes into account the inversion formula £ -1 = —£ / |£ |2 for vectors £ G R n C C£(Rn ). The second equation in (4.84) is analogous. □ Note that by multiplying (4.86) by fij —f from the right, we eventually arrive at (4 .87)

oaUi - f ) ~ \ f i j - fi) = OLj(fj - f ) - \ f i j - fj),

4. Special Classes of Discrete Surfaces

158

which is thus demonstrated to be equivalent to (4.85), a fact which is not obvious because of the noncommutativity. Finally, the diagonal flip / <-> fij turns (4.86) into (otx - otj)(fij - f ) 1 = o t i ( f i j - f i )

1 - aj i f i j - fi) \

and, comparing the latter equation with (4.86), one finds an important con­ sequence of (4.63): (4.88)

ai i f i - f ) 1 - otjifj - f ) 1 = otiifij - f j ) 1 - otjifij - ft)

Thus, we have found an alternative, noncommutative proof of the exactness of the one-form 5/*.

4.3.7. 3D consistency and zero curvature representation. We now turn to a direct algebraic proof of Theorem 4.26. This proof, like many other results on discrete isothermic nets, admits an immediate generalization to the case when the fields / in the cross-ratio equation (4.63) take values in an arbitrary associative algebra A with unit (over a field X) , with the labellings ai taking values in X. In our geometric situation A = Q£(RN ) and X = E. P roof of Theorem 4.26. We have to show that, imposing the cross-ratio equation on all six faces of an elementary cube C123 of Z3, the three values for fi23, coming from the faces TiCjk, coincide. I11 the case of commuting fields / this would be a result of a straightforward computation; however in the noncommutative context such a computation could hardly be performed without the aid of an ingenious matrix formalism. It is customary to repre­ sent Mobius transformations on C as a linear action of the group GL(2 , C); we extend this idea to define the action on A of the group of invertible 2 x 2 matrices with entries in A by the formula

This is easily seen to be indeed a left action of the group, provided the group multiplication is defined by the natural formula a' d

b' d!

a c

b d

We want to read equation (4.63) as giving fij in the form of a linear-fractional function of fj with coefficients depending on / , fi (and on ai, aj ). Towards this aim, we write the cross-ratio equation (4.63) as

4.3. Discrete isothermic nets

159

Representing the left-hand side as f i j —fj = (fij —fi) —(fj —fi), we transform the latter equation to the form ( k - fi) ( i + ^ ( / - /t ) _1( /, - / ) ) = f j - h ,

or, finally, fa

- f i

=

(if,

- /) + ( /-

fi))

(1 +

^

(/ -

fi)~ \fj

-

/))

1

The matrix form of this equation is (4.89)

fij - f i = L ( f , / , ati,aj)[fj - /],

where 1

(4.90)

f ~ f i \

L ( f , f,oti,otj) =

Thus, equation (4.63) on the faces C13, C23 of the elementary 3D cube C123 can be written as (4.91)

/l3 - fl

=

L ( f l , f, OL1 , 0 :3) [/3 - /],

(4.92)

f ‘23 ~~ f ‘2 =

L ( f 2 , / , OL2 , a 3)[/3 —/]•

From (4.91), (4.92) we derive, by the shift in the direction of the second, resp. first coordinate, the expressions for /123 obtained from the cross-ratio equation on the face T2 C1.3 , resp. on 7-1623(4.93)

/l23 —/l2

=

L ( f 1 2 , /2, «1, a3)[/23 —/2],

(4.94)

/l23 —/12

=

L ( f i 2 , f l , a 2, 0 3 )[/13 - fl}-

Substituting (4.91), (4.92) in the right-hand sides of (4.94), (4.93), respec­ tively, we represent the equality between these two values of /123 (which has to be demonstrated) in the following form: (4.95)

L ( / i 2 , / i , a 2,tt 3 ) i ( / i , / , a i , Q 3)[/3 - /] =

L( f i 2, f 2, at l , a 3 ) L( f 2 , f , a2, at 3 )[f3 - /].

We will show that actually a stronger claim holds: (4.96) L ( f 12, / 1, q 2, a 3 )L (/i, / , a i5 a 3) = L ( /i2, f i , c*i, a 3)L (/2, / , a 2, <*3)Indeed, the (12)-entries on both sides are equal to f — f \ 2. Equating the (11 gentries is equivalent to equation (4.85), equating the (22 )-entries is equivalent to the (inverted) equation (4.87), and equating the 21-entries is equivalent to equation (4.88). This finishes the proof. □

4. Special Classes of Discrete Surfaces

160

R em ark. The matrices L(fi , / , c^, ay) which appeared in this proof are ac­ tually an extremely important attribute of integrable (3D consistent) 2 D equations. They are known as transition matrices of a zero curvature repre­ sentation for such equations, and will be discussed in more detail in Chapter 6 . At this point, we only mention the following feature of (4.96): this equa­ tion is satisfied identically with respect to the parameter as, which is actu­ ally the only remainder of the third coordinate direction in this equation (all its other ingredients refer to the two-dimensional coordinate plane 12). This parameter is usually denoted by A and is known as the spectral parameter of the zero curvature representation. It is the dependence on the spectral parameter that allows for an application of powerful analytic methods of the theory of integrable systems. In the latter proof we demonstrated how to derive a zero curvature representation; actually, this derivation method is of a very general nature, since it is based on nothing more than the 3D consistency.

4.3.8. Continuous lim it. In order to enable a continuous limit to smooth isothermic surfaces, one should start with discrete isothermic surfaces (dis­ crete isothermic nets with m — 2 ) with embedded elementary quadrilaterals. It is convenient to represent their negative cross-ratios as (4.97)

= 012

with positive labels a \ and <22 • Formally, this means nothing more than changing the notation > —0L2 . This operation also puts some further expressions into the form which is closer to their continuous counterparts. We keep formula (4.67) as it stands; this implies a slight modification in the definition of the function s, namely s(u) 1—►(—l ) U2s(w), which also yields the similar modification in the lift s. This redefinition assures the positivity of s in the case of positive a\ , <22 . Equation (4.81) turns into (A OS'! (4.98)

/> ff* - n, * 01

12

\ Si fr

5

ssi

A $2 ff* - -n0,L2 -j,^ /. 12 l<*2f r

? ss2

which is a direct discrete analog of (1.73). Darboux transformations in the context of (4.97) take the form (4.99)

9( / , / i , / r , / +) = y

,

q ( f , f 2, t i , f +) = - y .

R em ark. Recall that in the smooth case (see Definition 1.28), the functions a i, OL2 can be absorbed into a reparametrization of the independent variables Ui 1—>
4.4. S-isothermic nets

by (4.97) with

ol\

161

= a2 =

1:

(4.100)

g ( / , / i , / i 2 , / 2) = - l .

This condition (all elementary quadrilaterals of / are conformal squares) may be regarded as a discretization of the conformality of the first funda­ mental form. Equation (4.100), being a particular case of (4.63) with a special labelling, enjoys all the properties of the general case. However, it is important to understand that it is not 3D consistent with itself\ i.e., it cannot be imposed on all faces of a 3D cube. Indeed, if a \ / a 2 = —1, then it is impossible to have additionally a 2/ a s = —1 and a \ / a s = —1 . The modification of s mentioned above makes it a discrete M-net in the light cone tA (4.101)

-

-

/

-

t \ t 2s + s = a \ 2 (r\s + r2s),

( s , t i s + t 2 s) a u = — — ----- — , \T\s, r 2 s)

and Darboux transformations can be formulated as discrete Moutard trans­ formations in Lj/V+1,1: (4.102)

n s + - s = bi (s+ - n s ) ,

r 2 s + + s = b2 (s+ + r 2 s).

4 .4 . S -is o th e r m ic n e ts According to Theorem 4.31, discrete isothermic nets are characterized as T-nets in the light cone L ^ +1,1 of the Minkowski space Replacing the light cone L ^ 1,1 by the hyperboloid L%+1'l = { Z £ R N+ 1'l :(ti , 0 = K2}

corresponds to blowing up points into spheres (see Section 9.3.2). As usual, we will only study in detail the N = 3 case, although the generalization for arbitrary N is straightforward.

D efinition 4.34. (S-isotherm ic net) A map (4.103)

S : Zm —> {oriented spheres in R3}

is called an S-isothermic net if the corresponding map s : Zm —>L V C R4,1,

(4.104)

J = ^ (c + e 0 + (|c |2 - r 2)eoo),

is a T-net in E4,1.

It follows from this definition that spheres of an S-isothermic net, con­ sidered as nonoriented spheres, build a special Q-congruence (cf. Definition 3.29).

4. Special Classes of Discrete Surfaces

162

Recall that sticking to the representatives (4.104) of oriented spheres in the Minkowski space R 4,1 of Mobius geometry is equivalent to consid­ ering their representatives in the space R 4,2 of Lie geometry with a fixed component along eg, K ^Lie — - (c + eo + (|c |2 —r 2)eoc, + reg). r

However, we will not use the Lie-geometric representatives in this section. An S-isothermic net is governed by the equation (4.105)

TiTjS -

S

= Clij {tj S - TiS)

in R4,1 with (4.106)

(s ,n s

-2 _

-

TjS)

(Tj Si T j § )

QLi -

O .J

K 1 ~ ( Ti S, T j S ) *

If the (signed) radii of all the spheres become uniformly small, r(u) ~ k s ( u ), k, —> 0 , then in the limit we recover a discrete isothermic net with metric s. Multidimensional consistency of T-nets in L ^1 (which is a particular case of Theorem 4.3) yields, in particular, Darboux transformations for Sisothermic nets, which are governed by equation (4.79). A Darboux trans­ form : Zm —> L'i-1 of a given S-isothermic net s : Zm —> L * 1 is uniquely specified by a choice of one of its spheres <s+ (0 ). The quantities c\t = (s,Tj.s) which have the meaning of cosines of the in­ tersection angles of the neighboring spheres, resp. of their so-called inversive distances if they do not intersect, possess the labelling property, i.e., depend on ui only. It turns out that this property almost characterizes S-isothermic nets among Q-congruences of spheres.

Theorem 4.35. (Labelling property yields M outard equation) Let four points s, s 2, S12 € span a three-dimensional vector space £ C R4,1 such that the restriction of the Minkowski scalar product to E is non­ degenerate, or, in other words , £ D E 1 = {0}. If (4.107)

( M l ) = («S2 ^ i 2) = Q.1,

( M 2) = ( SUS 12) = « 2 ,

then one of the two relations holds: a Moutard equation with minus signs ,

(4.108)

S\2 - s =

2a i , / >2 (.S-2 - Si), k2 - ( s 1, s 2)

or a Moutard equation with plus signs ,

(4.109)

sj 2 + s =

^ * ° 2. (si + s2). ^ + { s i , s 2)

163

4.4. S-isothermic nets

Proof. For simplicity of notations, we set in this proof

k

= 1 (the gen­

eral case is obtained by a simple re-scaling). Write the linear dependency condition as (4.110)

<§12 = As + /iSi + vs2-

There are three conditions for the three unknown coefficients X,ji,v. Two linear conditions are obtained from the requirements (4.107) by computing scalar products of (4.110) with 8i, 82- Denoting 7 = (51, 82), we put them as: (4.111)

\ a i + f i + i"y

=

« 2,

(4.112)

A«2 + HI + v

=

&i-

The third, quadratic, condition appears from (812, 812) = 1: (4.113)

A2 -f- fj2 -f- i/ 2 -f- 2A//qi + 2 Xi/a 2 + 2fiv^ = 1.

First assume that 7 2 ^ 1. Then the first two equations (4.111), (4.112) can be solved for /z, v in terms of A: ^ 11^ (4.I14J

_ a 2 - « i 7 - X(ai - a 27 ) fl-

l - 72

’

, _ a i “ a 27 - K a 2 ~ a 17) I - 72 *

V~

Upon substitution of expressions (4.114) into (4.113), one arrives at a qua­ dratic equation for A, which after massive simplifications turns into: (a 2 + a 2 + 7 2 - 2 a ia 27 - 1)(A2 - 1) == 0. If the numeric pre-factor in this quadratic equation does not vanish, that is, if a\ + a 2 + 7 2 — 2 a\a 2'y — 1 i=- 0, we find two solutions A = 1 and A = —1, which, being substituted into (4.114), yield ol\

— a2

i/ = - f x = —-------- , 1- 7

resp.

a\ + a 2

11 = 1/ = —

-------. 1+7

This corresponds to the Moutard equations (4.108), resp. (4.109). Equality (4.115)

a 2 + a 2 + 7 2 — 2 a \ a 2^ — 1 = 0

is interpreted as vanishing of the determinant of the system A + B a 1 + Cot2 — 0, A a 1 + B + C 7 = 0, Aa 2 + S 7 + C = 0,

for ( A ,B ,C ) , which is equivalent to (
164

4. Special Classes of Discrete Surfaces

(otherwise we would have the excluded case (4.115)). It follows that A = =pl and fi ± v — ct2 ± a\. Substituting this into (4.113) results in fi = ± u = (a 2 =t a i)/2 , so that one again obtains Moutard equations (4.109), resp. (4.108).

□

An interesting particular case of S-isothermic nets is characterized by touching of any pair of neighboring spheres. In this case the limit of small spheres is not relevant; therefore it is convenient to restrict considerations to a fixed value of k, say k = 1.

Theorem 4.36. (S-isothermic surface of pairwise tangent spheres) A Q-congruence of spheres in which any two neighboring spheres have an ex­ terior tangency is an S-isothermic surface (in particular, it is with necessity two-dimensional).

Proof. The touching condition means that all ai = (s, TiS) can, in principle, take values ± 1. Consider first the case m = 2 . Orienting all the spheres in the same way, for instance so that all normals point outside, we have (4.116)

( M i ) = ( M 2 ) = { s i , s n ) = ( s 2 , s n ) = - 1.

Now Theorem 4.35 yields that s, si, s2j s 12 satisfy the discrete Moutard equa­ tion with plus signs: (4.117) (since the version with minus signs becomes trivial for a\ = 0:2)- We know that the Moutard equation with plus signs is not 3D consistent. Of course, one can easily return to the original Definition 4.34 by orienting the spheres in every second row differently, so that they normals would point insides and there would hold ( s , S i ) = ( § 2 , S 12 ) =

- 1,

(s,

S2) =

(«i, S12)

=

1.

This change of orientation would lead to the discrete Moutard equation with minus signs. □ On a more general note, S-isothermic surfaces (i.e., S-isothermic nets with m = 2 ) are more conveniently described as M-nets in L ^ +1,1, governed by equation (4.109). As usual, the relation to the description as T-nets in L ^ +1,1, governed by equation (4.108), is established via the change of sign s (u i,u2) 1—►(—l ) U2s(ui, u2). In geometric terms, this amounts to the change of orientation (of oriented radius) of all the spheres along every second row in one of coordinate directions. Darboux transformations of S-isothermic surfaces are governed by equation (4.102). We now turn to geometric properties of S-isothermic nets.

165

4.4. S-isothermic nets

Theorem 4.37. (Centers and radii of an S-isothermic net) A map (4.103) is an S-isothermic net if and only if its nonoriented spheres build a Q-congruence, and the centers c : Zm —> M3 build a Konigs net with the signed radii r : Zm —>M* playing the role of the function v. In other words, for any elementary quadrilateral (c, Q ,Q j,C j ) , if M denotes the intersection point of its diagonals, then

/4 11 8 \ {

’

Tjj l(M ,c)

r ’

l(M , Cj)

Vj

l(M ,a )

ri'

Proof. This is a direct consequence of formula (4.104) and Theorem 2.32.

□

We can now give an alternative, more geometric, proof of Theorem 4.36. One has to prove relation (4.118) under the tangency condition for the hy­ perspheres S. Obviously, this situation belongs to case (i) of Q-congruences, so the four spheres S', Si, S 2, and S\2 of every elementary quadrilateral have a common orthogonal circle. Consider the section of the four spheres by the plane of their centers (where also the common orthogonal circle lies). Let

Figure 4.13. An elementary quadrilateral of an S-isothermic surface built of pairwise tangent spheres.

A , B , C , D be the tangency points of S with Si, of S with S2, of Si with

S 12, and of S2 with S 12, respectively. The common orthogonal circle is inscribed into the quadrilateral (c, ci, C12, c2) and touches its sides at the points A yJ5, C, D. The key observation is now the following property of quadrilaterals with an inscribed circle: the intersection point (AD) fl (BC) of the lines connecting the opposite tangency points coincides with M , the intersection point of the diagonals (CC12) and (C1C2). Actually, this is nothing but a degenerate case of Brianchon’s theorem, see Theorem 9.20. Consider

4. Special Classes of Discrete Surfaces

166

now two triangles A(c. A, M ) and A (ci 2, D, M ). By the sine law, sin Z(Ac, A M ) [d tfj

sin /.(M c , M A )

sin Z ( D c i 2 , D M ) ’

=

\c12M \

sin Z (M c\2, M D ) =

\c12D\

‘

But, obviously, Z(M c, M A ) = Z (M ci 2, M D ), Therefore we find:

l ( T c , A M ) = tt - l { D c ? 2, D M ).

|ci2M | = |ci2-P| = ryz

\cM\

\cA\

r '

This proves relations (4.118). In a similar manner one demonstrates the following geometric character­ ization of general S-isothermic nets of type (i) (with a common orthogonal circle for every elementary quadrilateral).

Theorem 4.38. (Geometry of S-isothermic nets) A map (4.103) is an S-isothermic net of type (i) if and only if its nonoriented spheres build a Q-congruence of type (i), and every elementary quadrilateral has the fol­ lowing property: if A, B are the points of intersection of the sphere S with the common orthogonal circle, and analogously for S{. S j, and Sij 7 then the intersection point (AAij) fl (B B ij) coincides with the intersection point

(.A iA j) fl (BiBj). This point coincides also with the intersection point of the diagonals M = (ccij) D (ciCj).

Figure 4.14. An elementary quadrilateral of a general S-isothermic net of type (i).

Clearly, the characterization of Theorem 4.38 can serve as a basis for construction of the fourth sphere Sij of an elementary quadrilateral of an S-isothermic net, if three spheres 5, Si, Sj are given: one finds first the point

4.4. S-isothermic nets

167

M = (A { A j ) H (B tB j ), then Aij and Bij as the intersection points of the orthogonal circle with ( AM) and with (B M ), respectively, and finally cij as the intersection point of the tangent lines to the orthogonal circle at and at Bjj. Note that this construction is actually not unique but rather two-fold, since it depends 011 the denomination of the points within each of the pairs { Ai . Bi } and { A j , B j } . Switching A t Bi corresponds to the change of orientation of the sphere Si.

For S-isothermic surfaces consisting of pairwise tangent spheres, one can give a nice geometric construction of the Christoffel dual. Towards this aim, one constructs quadrilaterals from the centers of the spheres, the centers of the orthogonal circles, and the touching points of the spheres. The resulting net / , called the central extension of s, is bipartite, with white vertices being the centers of the spheres and of the orthogonal circles, and with black vertices being the touching points. All elementary quadrilaterals of / are of the kite form with two right angles (those at the black vertices); see Figure 4.15. One easily sees that the converse statement is also true: a bipartite Q-net all of whose quadrilaterals are orthogonal kites (with right angles at the black vertices) is the central extension of an S-isothermic surface built of pairwise touching spheres.

Figure 4.15. Elementary quadrilaterals of the central extension of an S-isothermic surface with pairwise tangent spheres are orthogonal kites.

Orthogonal kites are conformal squares (see Exercise 4.9), which yields that the central extension / is a discrete isothermic net, with cross-ratios of all the elementary quadrilaterals q = —1.

Theorem 4.39. (Christoffel dual of the central extension of an Sisothermic surface) The Christoffel dual f* of the central extension f

168

4. Special Classes of Discrete Surfaces

consists of orthogonal kites and therefore is the central extension of a certain S-isothermic net S* : Z 2 —> {spheres in M3}, called the Christoffel dual of S. The radii r* : Z 2 —> M* of the spheres S* are given by r* = r 1, and the centers c* : Z 2 —>M3 satisfy

(4.119)

6 lC* = —

rr i

,

S2c* = - — . 7T2

Proof. The side lengths of the dual quadrilateral to a conformal square are inverse to the corresponding side lengths of the original quadrilateral, see part a) of Exercise 4.10. This proves r* — r ~ l . Considering two neighboring quadrilaterals, we see that the side [c*c*] is parallel to [cq], while their lengths are related by

- I + _L - r + Vi r

n

rn

\Ci - c\

re­

paying attention to the directions of the dual sides (see Exercise 4.10), this proves equation (4.119). □ Duality of S-isothermic nets built of pairwise touching spheres is illus­ trated in Figure 4.16.

Figure 4.16. Duality of S-isothermic surfaces built of pairwise touching spheres.

Remarkably, formulas of Theorem 4.39 allow one to construct Christoffel dual nets also for general S-isothermic nets. The following theorem can be considered as a generalization of Theorem 4.32.

Theorem 4.40. (Christoffel dual S-isothermic net) Let S : Zm —> {oriented spheres in M3}

169

4.4. S-isothermic nets

be an S-isothermic net. Denote the Euclidean centers and (signed) radii of S by c : Zm —>R3 and r : Zm —» R, respectively. Then the R3-valued discrete one-form 5c* defined by S'C SiC* = — , rn

(4.120)

1 < i < m,

is exact, so that its integration defines (up to a translation) a function c* :

Zm —>R3. Define also r * : Zm —>R by r* = r ~ l . Then the spheres S * with the centers c* and radii r* form an S-isothermic net, called Christoffel dual to S.

Proof. Consider equation (4.76), in terms of S = r ~l (c + e 0 + (|c|2 — r 2)eoo) • Its eo-part yields:

= (r^1 — r ~l ) / ( r j l — r ~ 1). This allows us to rewrite

(4.76) as (4.121)

(r j l - r j l)(Jij - s) = (r^1 - r-1 ) ^ - Sj).

A direct computation shows that the R -part of this equation can be rewrit­ ten as (4 1 2 2 )

cij ~ ci _

C1 ~ c /y*rr* .

cj — c

cij ~ C3

ry*/y .

rr* . rr* . .

ry* . rr* . .

Ill

\ ij

II j

1J1IJ

which is equivalent to the exactness of the form 5c*, defined by (4.120), on an elementary quadrilateral. In the same way, the e^-part of (4.121) is equivalent to the exactness of the discrete form Sw defined by

6i(\c\2 -

r 2)

OiW = ----------------- , rn

1 < %< m.

For similar reasons, the second claim of the theorem is equivalent to the exactness of the form * SiW* -

^(|c*|2 - ( r * ) 2) 11....1 ^.. ^ ... ... 1 , r ri

1 < i < m,

where, we recall, r* = 1jr. With the help of c* —c* = (c* —c)/rn , one easily checks that the forms 5w and 5w * can be written as S{W

=

(C* - C*,Ci + c) -

— + — ,

SiW*

=

(ci - c, c* + c*) - — + — .

r

n

n

r

The sum of these one-forms is exact: Si{w + w*) = 2(c*, Ci) - 2(c*, c);

therefore the exactness of one of these forms is equivalent to the exactness of the other. □

170

4. Special Classes of Discrete Surfaces

4.5. Discrete surfaces with constant curvature Here we present a curvature theory for polyhedral surfaces (discrete surfaces with planar faces) based on a discrete version of Steiner’s formula (1.55). We deal essentially with Q-nets, although some parts of the curvature the­ ory presented here can be applied to more general polyhedral surfaces, not necessarily quadrilateral. As in the smooth case, the curvatures are derived from the change in surface area as we move through a one-dimensional affine space of parallel surfaces. Discrete surfaces with constant curvature defined in this way possess nice geometric properties; in particular they are special discrete Koenigs nets.

4.5.1. Parallel discrete surfaces and line congruences. Having in mind Steiner’s formula (1.55) for the curvatures, we consider parallel polyhe­ dral surfaces defined as discrete surfaces with parallel corresponding edges. Polyhedral surfaces parallel to a given surface build a vector space. Given two parallel surfaces / and / + , the formula ft — t f + + (1 — t ) f gives an interpolating family of parallel surfaces. The difference surface n — / + — / is also parallel to both / + and / , and the above-mentioned one-parameter family of parallel surfaces can be seen as built from the surface / and its “generalized Gauss map” n: (4.123)

ft. = f + tn,

t e R.

Parallel discrete Q-nets are related by the Combescure transformation; see Definition 2.17. According to Theorem 2.18, in the case of Q-nets, family (4.123) of parallel discrete surfaces can be interpreted as a Q-net / : Z2 —> R3 with a line congruence £ : Z2 —> £ 3. The directions of the lines i are given by the generalized Gauss map n : Z 2 —> R3.

4.5.2. Polygons with parallel edges and mixed area. We start with a theory of polygons with parallel edges (recall that such polygons build faces of parallel surfaces). Let v i , . . . , v k € RP1 = Sx/ { ± / }

be a sequence of tangent directions of a k-gon P = (p i,. . . ,p /j in a plane; Pi+1 — Pi || Vi- Denote by !?('<;), = (vi , . . . ,!>&), the space of k-gons with edges parallel to u i ,...,t ^ . The polygons are not supposed to be convex nor embedded. They may have degenerated edges. CP(i;) is a A*-dimensional vector space. Factoring out translations (for example, normalizing pi = 0), we obtain a (k — 2)-dimensional vector space 3)(^).

4.5. Discrete surfaces with constant curvature

171

Let A (P ) be the oriented area of the polygon P. The oriented area of a k-gon with vertices p i ,. . . ,Pk,Ph+\ = pi is equal to

A(P) =

1 J*

^

j i—1

where [a, 6] = det(a, b) is the area form in the plane. For a quadrilateral p = (P1,P2,P3,P4) with oriented edges a = p2 - p i , b = p3 - p 2, c = p4 - p 3, d — pi —p4, we have (4.124)

A(P) = ^([a,&] + M ) .

The oriented area >1 is a quadratic form on the vector space Its corresponding bilinear symmetric form is of central importance for the following theory.

Definition 4.41. (Mixed area) Let P and Q be two k-gons with parallel corresponding (possibly degenerated) edges. Their mixed area is given by the bilinear symmetric form\ A(P, Q) = 1{- A { P + Q ) ~ A (P ) - A(Q )).

The area of a linear combination of two polygons is given by the qua­ dratic polynomial (4.125)

A (P + tQ) = A (P ) + 2 tA(P, Q) + t2A(Q).

We have a sort of scalar product j4(-, •) on the space parallel edges. It is natural to investigate which polygons with respect to the mixed area bilinear symmetric form. this way one obtains the dual quadrilaterals from Section

of polygons with are “orthogonal” It turns out that 2.3.1.

Theorem 4.42. (Dual quadrilaterals via mixed area) Two quadrilat­ erals P = (pi,P2,P3,P4) and Q — (gi,?2>93?94) with parallel corresponding edges, p*+i —pi ||qi+\ — qi, i G Z (mod 4), have vanishing mixed area A (P ,Q ) = 0 if and only if they are dual, i.e., if their noncorresponding diagonals are parallel: ( p m ) II (9294),

(P2P4) II (q m )-

Proof. Denote the edges of the quadrilaterals P and Q as in Figure 4.17. Formula (4.124) implies that the area of the quadrilateral P + tQ is given by A (P + tQ) = i([a + ta\ b + tb*] + [c + tc \ d + td*)).

4. Special Classes of Discrete Surfaces

172

Figure 4 .1 7 . Dual quadrilaterals.

Identifying the linear terms in t and using the identity a + b + c + d — 0, we get 4A {P ,Q )

=

[a,b*] + [a*,b] + [c,d*] + [c\d\

=

[ft + 6, 6*] + [a*, a + b] + [c + d, d ] + [c*, c + d]

=

[a+ 6, 6* - a* - d * + c*].

Vanishing of the last expression is equivalent to the parallelism of the non­ corresponding diagonals, (a + b) || (6* + c*). □ A quadrilateral with vanishing area, A (P ) = 0, is self-dual, A(P, P) — 0, and has parallel diagonals. In the case A (P ) ^ 0, the existence and uniqueness of the dual quadrilateral proven in Lemma 2.20 now follow from the fact that the corresponding space is two-dimensional. Moreover, a quadrilateral P with nonvanishing area and its dual P * build an orthogonal basis of the space ? (v ): (4.126)

A (\ P + nP*) = A2A (P ) + h2A (P *).

We call the space T(v ), v = (v\, V2 , ^3,^ 4) (and all the quadrilaterals in this space) nondegenerate if every two of its consecutive tangent directions are different, Vi+\ / The signature of the area form A : ? (v ) —> R depends on the quadruple v = ^2, ^3,^ 4) E (RP1)4 and can be also characterized in terms of the quadrilaterals in CP(v ).

Theorem 4.43. (Signature of the area form) Suppose y (v ), v — {vi,V 2 , vs, v±), is a nondegenerate space of quadrilaterals with consecutive edges parallel to the tangent directions i>i, t>2>^3>^4 € RP1 = S1/ { ± / } ? and let P € T(f) be a quadrilateral with nonvanishing edges.

The area form

A : CP(v) —>R is indefinite (resp. definite) if and only if two of the following equivalent conditions hold:

(i) the cross-ratio q(v 1, t>2, ^3, ^4) < 0 (resp. q > 0 ),

4.5. Discrete surfaces with constant curvature

173

(ii) all vertices of P are extremal points of their convex hull (resp. one of the vertices of P lies in the interior of the convex hull of the other three vertices). See Figure 4.18.

A (P ) > 0

A(P*) > 0

Figure 4.18. Left: The signature of the area form is indefinite; the vertices of the polygons lie on the boundary of their convex hull. Right: The signature of the area form is definite; for any quadrilateral of the family one vertex is in the interior of the convex hull of the other three.

The formulation of this theorem may require comments. The cross-ratio can be computed as , X

q {’

[ « l ,t ? 2 p 3 ,« 4 ]

[ h ,h } [ n ,v i V

where V{ £ R2 are some representatives of vt € RP1. Let us also mention that the property (ii) is independent of the choice of the quadrilateral P from the family. If A (P ) =/=- 0, the diagonal form (4.126) allows us also to characterize two alternative cases described in Theorem 4.43 in terms of the areas of P and of its dual P*: the area form is indefinite (resp. definite) if A {P )A (P *) < 0 (resp. A {P )A (P *) > 0).

Proof. The sign of the cross-ratio q(vi,v2, ^3, ^4) can be characterized com­ binatorially. The cross-ratio is positive if and only if the directions v\, ^3 G MP1 are separated by the directions v2,va G MP1 (recall that v\ ^ v 2 ^ ^3 7^ ^4 7^ ^i)> &nd q < 0 in the opposite case. It is not difficult to see that in the last case, when the pairs of directions v\,vs and ^2,^4 do not

intertwine, the edge lines can be realized by a convex quadrilateral. The whole corresponding family 7(v) consists of convex and of nonembedded quadrilaterals, and for both types the vertices lie on the boundary of their convex hull. Similarly one can check that when the pairs of directions v\, v% and ^2,^4 intertwine, the corresponding family 7(v) consists of embedded nonconvex quadrilaterals. □

4.5.3. Curvatures of a polyhedral surface with a parallel Gauss map. Consider a polyhedral surface / equipped with a congruence of lines £ such that every vertex has a line passing through it and the lines assigned

to adjacent vertices are coplanar. Our main example is a Q-net / : Z 2 —>M3

174

4. Special Classes of Discrete Surfaces

with a line congruence £ : Z 2 —►L 3 such that f ( u ) E £{u) for all u G Z 2. Let n : Z 2 —►E3 be the corresponding generalized Gauss map. If £ is simply connected, then the net n is determined up to a constant factor and is fixed as soon as the length of the normal at one vertex is prescribed.

Theorem 4.44. (Parallel surface area) The area of the parallel surface ft — / + tn obeys the law

(4.127)

(i - 2tH P + t 2K p ) A ( f(P ) ),

A (ft) = P

where

m „„,

„

<4-128)

H p=—

A (f(P ) ,n (P ) ) A im )

A (n {P )) '

K p = m p y y

Here the sum is taken over all (combinatorial) faces P , and f ( P ) and n(P) are the corresponding faces of the surface f and its generalized Gauss map n.

Proof. Since the corresponding faces and edges of discrete surfaces / and n are parallel, the claim follows from formula (4.125).

□

Having in mind Steiner’s formula, we come to the following natural def­ inition of the curvatures in the discrete case.

Definition 4.45. (Mean and Gaussian curvatures of polyhedral sur­ faces) Let ( /, n) be two parallel polyhedral surfaces. We consider n as the generalized Gauss map of f . The functions Hp and K p on the faces given by (4.128) are the mean and the Gaussian curvatures of the pair ( /, n), i.e., of the polyhedral surface f with respect to the Gauss map n.

Note that, as in the smooth case, the Gaussian curvature is defined as the quotient of the areas of the Gauss image and of the original surface. Since for a given polyhedral surface with a line congruence the map n is defined up to a common factor, the curvatures at the faces are also defined up to multiplication by a constant. The principal curvatures k i,/^ at the faces are naturally defined using the formulas H — (k i + k2) /2 and K = k\k2 as the zeros of the quadratic polynomial

(4.129)

A ( f t) = (1 - 2tH + t 2K ) A ( f ) = (1 - i« i)(l - tK2)A (f).

Definition 4.46. (Principal curvatures of Q-nets) Let ( /, n) : Z 2 —► R3 x M3 be a Q-net with a generalized Gauss map. Assume that the area forms A : 7 —> E are indefinite for all the faces f ( P ) . Then the functions ^1)^2 o f (4.129) are real-valued and are called the principal curvatures of the pair ( /, n).

4.5. Discrete surfaces with constant curvature

175

The results of Section 4.5.2 imply that the principal curvatures exist for quadrilateral faces with vertices on the boundary of the convex hull. In particular, for a circular net, principal curvatures exist for any Gauss map.

4.5.4. Q-nets with constant curvature. Let ( /, n) : Z2 —> R3+3 be a Q-net with a generalized Gauss map, and let ft = f + t n : Z 2 —> M3, t E M be the corresponding one parameter family of parallel Q-nets. We define special classes of surfaces as in the classical surface theory, the only difference is that the Gauss map is not determined by the surface. The treatment is similar to the approach in relative differential geometry.

Definition 4.47. (Q-nets with constant curvature) We say that a pair ( /, n) has constant mean (resp.

Gaussian) curvature if the mean (resp. Gaussian) curvatures defined by (4.128) for all faces of the Q-net are equal.

If the m,ean curvature vanishes identically, H = 0, then the pair (/, n) is called minimal.

Although this definition refers to the Gauss map, the normalization of the length of n is irrelevant, and the notion of constant curvature nets is well defined for Q-nets equipped with line congruences.

Theorem 4.48. (Minimal Q-net with a line congruence) A pair ( /, n) is minim,al if and only if f : Z 2 —» M3 is a discrete Koenigs net and n : Z 2 —> M3 is its Christoffel dual n = f * .

Proof. We have the equivalence H = 0 &

A { f,n ) = 0 4^ n = f*.

□

F igure 4 .1 9 . Discrete Koenigs nets interpreted as a Gauss image n, and its Christoffel dual minimal Q-net / = n *.

This result is analogous to the classical Theorem 1.36. Figure 4.19 presents an example of a minimal Q-net constructed as the Christoffel dual of its Gauss image n, which is a discrete Koenigs net. The statement about surfaces with nonvanishing constant mean curva­ ture resembles the corresponding facts of the classical theory; see Theorem 1.37. *

176

4. Special Classes of Discrete Surfaces

Theorem 4.49. (Constant mean curvature Q-net with a line con­ gruence) A pair (/, n) has constant mean curvature Hq if and only if /

: Z 2 —> M3 is a discrete Koenigs net and the parallel surface f i / H 0 is

the Christoffel dual of f : r

= / + jU .

The mean curvature of this parallel surface ( / + jj^n, —n) (with the reversed Gauss map) is also constant and equal to H q. The mid-surface f + ^j^n has constant positive Gaussian curvature K q = 4 H q with respect to the same Gauss map n.

Proof. We have the equivalence A ( f , n ) = - H 0A ( f ) &

A ( f , f + jr n ) = 0 O

/* = / + ±-n .

For the Gaussian curvature of the mid-surface we get =

m

A(n)

= _____________ A(n) _____________ =

A ( / + 2 S o n)

A( f +

+ W l A (n)

2

°'

It turns out that all surfaces parallel to a surface with constant curvature have remarkable curvature properties, in complete analogy to the classical surface theory; cf. Theorem 1.37.

Theorem 4.50. (Parallel of constant mean curvature Q-nets are linear Weingarten) Let ( /, n) be a pair of Q-nets with constant mean curvature. Consider the family of parallel Q-nets ft = f + tn. Then for any t the pair (f t , n) is linear Weingarten, i.e., its mean and Gaussian curvatures Ht and Kt satisfy a linear relation

(4.130)

aHt + 0 K t = 1

with constant coefficients a,/3.

Proof. This can be checked by a direct computation; see Exercise 4.20.

□

We see that any discrete Koenigs net / can be extended to a minimal or to a constant mean curvature Q-net by an appropriate choice of the Gauss map n. Indeed, ( /, n) is minimal for n = /* ; ( /, n) has constant mean curvature for n = /* — f . However, n defined in such generality can lead us too far away from the smooth theory. It is natural to look for additional requirements which bring it closer to the Gauss map of a surface. In the smooth case, n is a map to

4.5. Discrete surfaces with constant curvature

177

the unit sphere. The following three discrete versions of this fact are natural to consider: (1) n is a polyhedral surface with all vertices on the unit sphere § 2. This implies that all faces of the Q-net n are circular. This con­ dition holds also for any parallel surface. In particular, / is also a circular net. (2) n is a polyhedral surface with all faces touching the unit sphere § 2. This implies that for any vertex p there is a cone of revolution with the tip p touching all faces of n incident to p. This property holds true for any parallel surface. In particular, / is also a conical net. (3) n is a polyhedral surface with all edges touching the unit sphere § 2. Polyhedra with this property are called Koebe polyhedra. For any vertex p, all edges incident to p lie on a cone of revolution with the tip p. Also this property holds true for any parallel surface, in particular for f . Such nets are called nets of Koebe type. The implementation of these additional requirements into the theory makes it more intriguing.

4.5.5. Curvature of principal contact element nets. In this section we are dealing with the case when the Gauss image n lies in the two-sphere S2, i.e., is unitary, \n\ = 1. It turns out that this is the case of principal contact element nets. Let / : Z 2 —►R3 be a Q-net with a parallel unit Gauss map n : Z 2 -► § 2. Introducing the oriented planes P which contain the corresponding vertices of / and are orthogonal to the corresponding normals n, we obtain a contact element net ( /, P) : Z 2 —> {contact elements in R3}; see Section 3.5. It can be canonically identified with the map ( /, n) : Z 2 -» R3 x S2 considered previously. We will call the latter also a contact element net.

Theorem 4.51. (Q-nets with unit Gauss maps are principal contact elements) Let f : 1? —> R3 be a Q-net with a parallel unit Gauss map n : Z 2 —> § 2. Then f is circular, and

(/,n ) : Z 2 —* R3 x S2 is a principal contact element net. Conversely, for a principal contact ele­ ment net ( f ,P ) , the net f is circular and the unit net n canonically corre­ sponding to P via n _L P , is a parallel Gauss map of f .

178

4. Special Classes of Discrete Surfaces

Proof. The circularity of / follows from the simple fact that any quadri­ lateral with edges parallel to the edges of a circular quadrilateral is also circular. Consider an elementary cube built by two parallel quadrilaterals of the nets / and / + n. All the side faces of this cube are trapezia, which implies that the contact element net ( /, n) is principal. □

Figure 4 .2 0 . Parallel Q-nets / and / + n with the unit Gauss map n. All the nets are circular. The pair ( / , n) build a principal contact element net.

The mean and the Gauss curvatures of the principal contact element nets ( /, n) are defined by formulas (4.128). In Section 4.5.3 it is shown that in the circular case the principal curvatures always exist.

4.5.6. Circular minimal nets and nets with constant mean curva­ ture. Minimal and constant mean curvature principal contact element nets are defined as in Section 4.5.4. Since circular Koenigs nets are isothermic nets, we obtain the following results from Theorems 4.48, 4.49.

Corollary 4.52. (Minimal circular surfaces as principal contact el­ ement nets) A principal contact element net ( /, n) : Z 2 —*■ R3 x § 2 is minimal if and only if the net n : Z 2 —> § 2 is isothermic and f — n* is its Christoffel dual.

Discrete isothermic nets are described through the cross-ratios of their quadrilaterals. The stereographic projection a : R2 —> § 2 preserves the cross-ratios. Combined with the formula / = n*, this yields a Weierstrass representation for discrete minimal surfaces through isothermic nets in a plane. See Exercise 4.21.

Corollary 4.53. (Circular surfaces with constant mean curvature as principal contact element nets) A principal contact element net (f. n ) : Z2 —» R3 x § 2 has constant mean curvature Ho ^ 0 if and only if the circular net f : Z2 —> R3 is isothermic and there exists its dual discrete isothermic surface /* : Z2 —> R3 at constant distance \f — f*\ — jj~. The unit Gauss

4.6. Exercises

179

map n which determines the principal contact element net ( /, n) is given by

(4.131)

n =

~ /)•

H 0(f*

The principal contact element net of the parallel surface ( f + has constant mean curvature H q. The mid-surface ( f +

—n) also

77^ n) has constant

GaxLSsian curvature 4 H q .

Proof. Only the “if” part of the claim may require some additional consid­ eration. If the discrete isothermic surfaces / and /* are at constant distance 1/ Hq, then the map n defined by (4.131) is unitary and thus circular. Again, as in the proof of Theorem 4.51, this implies that the contact element net ( /, n) is principal. Its mean curvature is given by A (/,n )

A ( f ,H 0 (f* - f) )

A (fJ )

A (fJ )

H 0.

□

Remark. We mention that there exists a theory of discrete minimal and constant mean curvature surfaces of Koebe type with discrete analogues of many famous classical surfaces. Its presentation lies beyond the scope of this book. 4.6. Exercises 4.1. Check the 3D consistency of the 2D equation (4.28). 4.2. Check by a direct computation the claim of Theorem 4.14. 4.3.* Work out the details of the proof of Theorem 4.15. 4.4.* Consider a Chebyshev quadrilateral ( / , / 1, / 12, jfe)- Let

and ip* be its internal angles. Furthermore, let (n ,n i, 77,12, n2) be the Lelieuvre (unit) normals satisfying the discrete Moutard equation with plus signs: 77-12 + n || n 1 + 722- Let (n,ni) = (77,2, 7742) = cosai and (n,n 2) = (^ 1, 77,12) — cos 0 2 . Suppose that ot\ and are chosen so that |tan(ai/2)| < 1 and |tan(a!2/2)| < 1. Show that the angles of the quadrilateral ( /, / 1, / 12, f 2) satisfy an involutive relation i * 1 - ne1^ elip = ---------— K— where

k

•

et(p =

1 - Ke%{p* k

— eiv>

= tan(ai/2) tan(a 2/ 2 ).

4.5. Show that the Backlund transformation (4.45), (4.46) of the trivial solution 4>(u) = 0 is given by the formula 0 + ('Ui, u2)

(c + a i \ wi f l + ca2 \ u2

tan — ^-7-— 1 = --------4 Vc \c — a\J a\ /

---------Vl VI — ca ca2J 2

where aj — tan(ay/2) and c = tan(7 / 2 ).

0 + (O,O)

tan — ^— - ,

4. Special Classes of Discrete Surfaces

180

4.6.* There exists a remarkable analytical device allowing us to construct a discrete K-surface / : Z 2 —* M3 from a solution 0 : Z 2 —►R of the Hirota equation. First the so-called extended frame \I/(-,A) : Z 2 —»•SU(2 )[A, A-1 ] is constructed by virtue of the compatible equations (4.132)

V(u + ej, A) = Uj(u , A)®(u, A),

j = 1, 2 .

These equations should be compared with equations (4.14) for the frame <E>. The compatibility of (4.132) is guaranteed by the zero curvature relation. Prove that the so-called Sym formula (4.133)

f x(u) = 2 A fr- 1( ^ A ) d * ^ ,A)

defines, for A G R, a family of discrete K-surfaces f\ : Z 2 —> su(2) ~ E3, which share one and the same solution 0 of the Hirota equation (the so-called associated family). For A = 1 this is the original K-surface / .

4.7. The extended frame of the solution from Exercise 4.5 equals, up to a scalar factor, tf(u,A) = where t0/ *

^

(

1

—i\ a i\ Ul (

1

)

1

(iA ->a2

iA -^ s V 2 1 )

is the extended frame of the trivial solution <j>{u) = 0 , and ei
-i\ c

_ iXc _ iih+

e- ^ +(«)/2

corresponds to the Backlund transformation. Applying Sym formula from Exercise 4.6, compute the corresponding discrete K-surface (Backlund trans­ formation of a straight line). For certain values of a\ = and for c = 1 the resulting surface is a discrete pseudosphere.

4.8. How can one construct a discrete K-surface with two straight asymp­ totic lines? Hint: The corresponding curves of the Gauss map must lie on two great circles of § 2. Such surfaces are known under the name of Amsler surfaces.

4.9. A conformal square is, by definition, a Mobius image of a square. Prove that a quadrilateral is a conformal square if and only if the cross-ratio of its sides is q = —1.

4.6. Exercises

181

4.10. a) Consider a conformal square in C with sides a , 6,c, d. Show that there exists a quadrilateral with the edges *

1

a_

,»

a ~ a~~ \a\2 ’

1 _____b_ b~

\b\2 ’

C _ c _ |c|2 ’

/ * _ _ ! _ ___ d~ |d|2 ’

that is, a* + 6* + c* + d* = 0 , and that this quadrilateral is dual to the original one. b) The same statement holds if the original quadrilateral is circular, with the real cross-ratio nc cy q = q{a,b,c,d) = — =

Q ,j3eR *,

and ci

* _

*

1 _ (y. _

a

1

ot |

a

.rt}

|ap C

c = a d = a w 2’

u* b

7*

[j —

b

^1

-

a b (j i , | o i

|op

n d

^2 = ^ W

4.11.* Given a discrete A-net, prove that its Gauss map (built from the unit normals, which must be distinguished from the Lelieuvre normals) is circular if and only if it is discrete isothermic.

4.12.* Consider the following specialization of conditions of the Miquel the­ orem (Theorem 9.21): let the points f i j E ( f i f j ) on the sides of the triangle A ( / i , / 2,/s ) be chosen collinear (Menelaus’ configuration). Prove that the intersection point /123 of the three circles through ( f i , f i j , f i k ) lies on the circumcircle of A ( f \ , f 2 , f 3 )] see Figure 4.21.

Figure 4.21. Miquel theorem for a Menelaus’ configuration.

182

4. Special Classes of Discrete Surfaces

4.13. Let / : Z2 —> WN be a discrete isothermic surface described by the cross-ratio equation g ( / , /i ,/ i 2 ,/ 2 ) = — • OL2

The transition matrices of its zero curvature representation are i /- /. a oLiU - f i r 1 i Define the moving frame ^ : Z 2 —> GL(2, C^(R^))[A] as follows: let ^(0, A) 1 - / ( I0 ) .; if ci,. . . , en is a sequence of edges connecting 0 — (0,0) with P u — («i,tt 2) £ Z 2, then

¥(u,A) =

f]

L (e- W O , A).

l< i< n

One now defines the so-called Calapso transform g = T\f : Z 2 —» R N by

°/ “ V

1

/

Here equivalence of 2 x 1 matrices with entries from C£(MN) is understood modulo simultaneous right multiplication of its entries with one and the same invertible element of the algebra, so that ( ^ ~ A = 0 one gets ^(u , 0) = ^

^

J , so that ^

V Clearly, for (-U, 0)

~ y

i

j-

Therefore, for small A, T \f = / + O(A). Prove that the Calapso transform is a discrete isothermic surface with the cross-ratios a i / ( l — Aai) 9 (9 , 91, 912, 92) = Q2/(1 _ Aa2) ■

4.14. Prove that the Calapso transformation g = T\f acts on all stars of the net / by Mobius transformations. That is, for every u G Z 2 there exists a Mobius transformation M\(u) : R N —►R N such that g(u) = M\(u) •f(u )

and

gl(u) = M\(u) •fi{u)

(i = ± 1 , ±2).

4.15. Given three pairwise touching spheres, prove that the circle through the touching points is orthogonal to all three spheres. 4.16. Given four pairwise (cyclically) touching spheres, prove that the four touching points are concircular, and that the circle intersects all spheres under equal angles. 4.17. In the previous exercise: if the circle intersects the spheres orthogo­ nally, then the spheres are linearly dependent (hence S-isothermic).

4.7. Bibliographical notes

183

4.18. Prove the so-called “Touching coins lemma” : whenever four circles in 3-space touch cyclically but do not lie on a common sphere, they intersect the sphere which passes through the points of contact orthogonally.

4.19. Complete the details in the proof of Theorem 4.43. 4.20.* Prove Theorem 4.50 and compute the coefficients a,/3. 4.21. Derive the following discrete Weierstrass representation for circular minimal surfaces: S if

=

A - gig i(l + gig) gi + g" g v - g

1-

g i - g

<12(1 i ( l + g 2 g )

g i - g

52-5

51 -

g

g-2 + g in -

g,

where g : Z 2 —» C is a discrete holoinorphic map, i.e., a solution of the cross-ratio equation

<7(5.51.512.52) = — ; C*2

see Chapter 8. Hint: The isothermic Gauss map n : Z 2 —> § 2 is given by the stereographic projection of g, 29

|y|- - 1

( n , + m 2 ' " 3 ) - ' | 9 p + r i 9 |2 + i.

4.22. Derive an explicit formula for the discrete Enneper minimal surface via the discrete Weierstrass representation of Section 4.5.5 by applying it to the standard square grid. The latter is the simplest isothermic net in a plane.

4.7. Bibliographical notes Section 4.1: Discrete Moutard nets in quadrics. General Moutard nets in quadrics were introduced in Bobenko-Suris (2005) (the first online version of this book), along with the most prominent example of the Moutard representatives of discrete isothermic nets. The latter example was gener­ alized in Bobenko-Suris (2007b), where discrete isothermic nets in various sphere geometries were investigated. Later the same class of nets (Koenigs nets in quadrics) was treated in Doliwa (2007b). Section 4.2:

Discrete K-nets.

The notion of discrete K-nets is due to Sauer (1950) in the case m — 2 and to Wunderlich (1951) in the case rn = 3. A study of discrete K-surfaces within the framework of the theory of integrable systems was performed in Bobenko-Pinkall (1996a) geometrically, and in Bobenko-Pinkall (1999) analytically. A presentation in BobenkoMatthes-Suris (2005) is based on the notion of consistency. The study of the Gauss map of K-surfaces leads to the notion of discrete Lorentz-harmonic

184

4. Special Classes of Discrete Surfaces

nets in § 2, also introduced in Bobenko-Pinkall (1996a), where the m — 2 case of Theorem 4.8 was first observed. Special classes of discrete K-surfaces were constructed by Hoffmann (1999) (discrete Arnsler surfaces; see also Exercise 4.8) and Pinkall (2008) (discrete K-cylinders that touch a plane along a closed curve and those ex­ hibiting a cone point). Color images of discrete K-surfaces are included in the book Bobenko-Seiler (1999). Discrete surfaces in Figures 4.3, 4.5 were produced using a software implementation by Ulrich Pinkall. K-surfaces are reciprocal parallel to geodesic conjugate nets, called Voss surfaces. Discrete Voss surfaces were introduced in Sauer-Graf (1931) as Q-surfaces with the property that the opposite angles at each vertex are equal. Sauer (1950) has shown that the relation between Voss surfaces and K-surfaces is preserved in the discrete setup; see also a modern presentation in Schief-Bobenko-Hoffmann (2008). The study of the angle between the asymptotic lines on discrete Ksurfaces leads to the discretization of the sine-Gordon equation, performed in Bobenko-Pinkall (1996a). The closely related integrable discretization of the sine-Gordon equation was derived by Hirota (1977b) without geomet­ ric interpretation. Its symplectic structure was studied in Faddeev-Volkov (1994). Stationary solutions of the discrete sine-Gordon equation describe a discrete pendulum, which was studied in Suris (1989), Bobenko-Kutz-Pinkall (1993). Besides discrete K-surfaces, there exist further remarkable special classes of discrete asymptotic nets: discrete affine spheres studied in Bobenko-Schief (1999a, b), and discrete Bianchi surfaces studied in Doliwa-NieszporskiSantini (2001). Section 4.3: Discrete isothermic nets. were introduced in Bobenko-Pinkall (1996b).

Discrete isothermic surfaces

Darboux transformations for discrete isothermic surfaces were intro­ duced in Hertrich-Jeromin-Hoffmann-Pinkall (1999). In particular, Theo­ rem 4.26 on the 3D consistency of the cross-ratio equation was given in this paper for the quaternionic cases TV = 3,4 under the name “hexa­ hedron lemma” with a computer algebra proof. An analytic description of the Darboux transformation as a dressing transformation was given in Ciesliriski (1999). Three-dimensional discrete isothermic nets were intro­ duced in Bobenko (1999) and Bobenko-Pinkall (1999). A conceptual proof of the 3D consistency in a more general context of an arbitrary associative algebra was given in Bobenko-Suris (2002b).

4.7. Bibliographical notes

185

The Calapso transformation for discrete isothermic surfaces (see Exer­ cises 4.13, 4.14) as well as permutability properties of various transforma­ tions are due to Hertrich-Jeromin (2000, 2003). Discrete isothermic surfaces in higher codimensions were studied by Schief (2001). Besides the discrete isothermic nets, there exists another interesting spe­ cial class of multidimensional circular nets. These are discrete analogs of Egorov metrics. They are characterized by the property that any elemen­ tary quadrilateral ( /, f i , f i j , fj) has two right angles at the vertices fi and fj. (Note that for this definition it is essential to fix the directions of all coordi­ nate axes.) The theory of discrete Egorov nets is due to Schief, AkhmetshinVorvovskij-Krichever (1999) and Doliwa-Santini (2000). Section 4.4: S-isothermic surfaces. The presentation of this section essentially follows Bobenko-Suris (2007b). S-isothermic surfaces, along with their dual surfaces were originally introduced in Bobenko-Pinkall (1999) for the special case of touching spheres. The general class of Definition 4.34, together with Darboux transformations and dual surfaces, is due to Hoffmann. Section 4.5: Discrete surfaces with constant curvature. Circular minimal surfaces were introduced in Bobenko-Pinkall (1996b) as Christof­ fel duals of their isothermic Gauss maps. The discrete Weierstrass rep­ resentation was also derived in this paper. Circular surfaces with con­ stant mean curvature appeared for the first time in Bobenko-Pinkall (1999) and Hertrich-Jeromin-Hoffmann-Pinkall (1999) as isothermic nets with a Christoffel dual at constant distance. In the second paper it was shown that equivalently circular surfaces with constant mean curvature can be defined as isothermic surfaces with a Darboux transform at constant distance. Curvatures of circular surfaces with respect to arbitrary Gauss maps n E S2 based on Steiner’s formula were introduced in Schief (2003a, 2006),

where it was also shown that the surfaces parallel to a surface with constant Gaussian curvature are linear Weingarten. A curvature theory for general polyhedral surfaces based on the notions of parallel surfaces and mixed area is developed in Pottmann-Liu-Wallner-Bobenko-Wang (2007) and BobenkoPottmann-Wallner (2008). In the circular case this theory yields the same class of surfaces with constant curvatures as originally defined in BobenkoPinkall (1999), Hertrich-Jeromin-Hoffmann-Pinkall (1999); see Corollaries 4.52, 4.53. Discrete surfaces in Figure 4.19 were produced using a software implementation by Peter Schroder. The theory of minimal surfaces of Koebe type was developed in BobenkoHoffmann-Springborn (2006). These surfaces are S-isothermic and their Gauss maps are Koebe polyhedra. Global results in this theory are based

4. Special Classes of Discrete Surfaces

186

on the remarkable fact that a Koebe polyhedron is essentially uniquely de­ termined by its combinatorics. This theory is closely related to the theory of orthogonal circle patterns (see Chapter 8 ). Section 4.6: Exercises. Ex. 4.6, 4.7: See Bobenko-Pinkall (1996a). Ex. 4.8: See Bobenko-Pinkall (1996a), Hoffmann (1999). Ex. 4.11: See Wallner-Pottmann (2008); the corresponding theorem for smooth surfaces can be found in Darboux (1914-27, §874). Ex. 4.13, 4.14: See Hertrich-Jeromin (2000, 2003). Ex. 4.18: See Bobenko-Hoffmann-Springborn (2006). Ex. 4.21, 4.22: See Bobenko-Pinkall (1996b).

Chapter 5

Approximation

We have already had several occasions to mention that the notions, construc­ tions and results of discrete differential geometry have not just qualitative similarity with their much more sophisticated counterparts in the smooth theory. Rather, the latter can be obtained from the former through a wellestablished continuous limit. Strictly speaking, such a continuous limit has been established up to now only for those geometries which are described by hyperbolic systems of difference, resp. differential, equations. It is this class of equations for which a rather detailed approximation theory can be developed, which is similar to the corresponding theory for ordinary differ­ ence and differential equations. Actually, this hyperbolic theory covers a substantial part of the nets considered in this book.

5.1. Discrete hyperbolic system s To formulate the general scheme that covers the majority of situations en­ countered so far, we will put our hyperbolic systems into the first order form. It should be stressed that this is necessary only for general theoretical considerations, and will never be done for concrete examples. Definition 5.1. (Hyperbolic system) A hyperbolic system of first order partial difference equations is a system of the form

(5.1) for functions

SiXk = flfc,i(z),

: Z M —>

i G £*,

with values in Banach spaces X k. For each

Xk, equations (5.1) are posed for i G £& C { 1 , . . . , M }, the nonempty set of

evolution directions of x k. The complement of static directions of x k.

— {!> •••>M } \

consists

187

188

5. Approximation

We think of the variable x k(u ) as attached to the elementary cell Gk(u) of dimension # 8 ^ adjacent to the point u G Z M and parallel to Sfc = Qk(u) =

{u E +

^

: Vi e [o, 1] j .

i€Sfc

Here, we recall, = {wG Z M : Ui = 0 if i £ 8 }, for an index set S

C

{ 1 ,..., M }.

Definition 5.2. (Goursat problem) 1) A local Goursat problem for the hyperbolic system (5.1) consists in finding a solution x k for all k and for all cells Ck within the elementary cube of Z M at the origin from the prescribed values x k(0). The system (5.1) is called consistent if the local Goursat problem for this system is uniquely solvable for arbitrary initial data a?fc(0 ). 2 ) A global Goursat problem consists in finding a solution of (5.1) on Z M subject to the following initial data:

(5.2)

Xk\3 &k=

where X k : 23s*. —> *Xk are given functions.

The following rather obvious but extremely important statement holds:

Theorem 5.3. (Well-posed Goursat problem) A Goursat problem for a consistent hyperbolic system (5.1) has a unique solution x on all o f Z M .

Consistency conditions read: 5jSiXk = Si$jXk for all i / j. Substituting (5.1), one gets the following equations: (5.3)

Sjgk,i(x) = 5igkj ( x ) ,

i ^ j,

or gk,i(x + gj(x)) -flfc,»(x) = gkj ( x + gi(x)) - g kJ(x), where g{(x) is a vector function whose £-th component is equal to gej(x) if i G £^, and is undefined otherwise.

Lemma 5.4. For a consistent system of hyperbolic equations (5.1), the func­ tion gke C Sk U {i}.

Proof. Equations (5.3) must hold identically in x. This implies that the function gk^ can only depend on those components X£ for which SjX£ is defined, i.e., for which j G £^. As (5.3) has to be satisfied for all j G £&, j 7^ z, one obtains that t k \ {z} C £^ for these t. □ It follows from Lemma 5.4 that for any subset S C { 1 , . . . , M }, equations of (5.1) for k with C § and for i G § form a closed subsystem, in the sense that gk^ depend on xa with S ^ c S only.

189

5.1. Discrete hyperbolic systems

Definition 5.5. (Essential dimension) The number (5.4)

d — 1 + max (#Sfc) k

is called the essential dimension of system (5.1).

If d — M , system (5.1) has no lower-dimensional hyperbolic subsystems. If d < M , then d-dimensional subsystems corresponding to S with #S = d are hyperbolic. In this case, consistency of system (5.1) is a manifestation of a very special property of its d-dimensional subsystems, which we treat as the discrete integrability (at least if one excludes certain noninteresting situations, such as trivial evolution in some of the directions). Section 6 will be devoted to an extensive treatment of integrability understood as consistency.

Example 1. Consider a difference equation with M — 3 independent vari­ ables: (5.5)

SiS2S$x = F (x , Six, S2x, Ssx, SiS2x , SiS^x, S2S^x).

One can pose a Goursat problem by prescribing the values of x on the coor­ dinate planes S 12, ® i 3, ®23- Equation (5.5) can be rewritten as a hyperbolic system of first order equations by introducing auxiliary dependent variables b, c, f , g, h:

(5.6)

r Six = a, S2a = / , < Sib = / ,

S2x = 6,

S3X = c,

53a = 3 , S3b = h,

8 ic = g, S2c = h , , Ssf = S2g = Sih = F (x , a, 6, c, / , 3 , /i).

It is natural to assume that the variable x lives on the points of the cubic lattice u G Z 3; the variables a, 6, c live on the edges Qi(u), Q2 (u), C3(u) of the lattice adjacent to the points u and parallel to the coordinate axes S 1, ® 2, ® 3>respectively; and the variables / , g, h are associated to two-cells (elementary squares) Ci2 (u ), Qis(u), C23(u) adjacent to the points u and par­ allel to the coordinate planes B 12, £ 13, ® 23, respectively. Thus, x has no sta­ tionary directions; the stationary directions of a, 6, c are {1}, { 2 }, {3}, while the stationary directions of f ,g ,h are { 1, 2 }, { 1, 3}, { 2 ,3 }, respectively. A Goursat problem for this system would be posed by prescribing the values of x at the point (0 , 0 , 0 ), the values of a, 6, c on the axes 231, fB 2, B 3, respec­ tively, and the values of / , g , h on the planes ® i 2, ® i 3, £ 23, respectively. The essential dimension of this discrete system is d = 3 = M . It is instruc­ tive to compare this construction with its continuous counterpart: equation (5 .5 ) is a natural discretization of the partial differential equation (5.7)

did2dsx = F (x , dix, d2x , d^x, did 2x , d ^ x , d2dsx).

5. Approximation

190

For the latter equation one can introduce auxiliary variables a,b,c, f ,g ,h via partial derivatives analogously to (5.6). All these variables would be on an equal footing, being defined just at the points u £ M3. Example 2. Consider the difference equations which govern M-dimensional Q-nets: (5.8)

SiSjX — CijdjX + CjiSiX.

Upon introducing auxiliary variables Vi they can be written as a hyperbolic system of first-order equations:

(5.9) ik

{Ticjk)ciki

^7^ j ^ k ^ 1.

The last equation is (2.7) from Section 2.1.1, where one can also find details about its origin, as well as about how one can put it in the form with the right-hand side depending only on the unshifted variables c^ . It is natural to assume that the variable x lives at the points of the cubic lattice u £ Z M; the variables Vi live at the edges 6 i(u) of the lattice adjacent to the points u and parallel to the coordinate axes 2?*, and the variables c^, cji for i < j live oat two-cells (elementary squares) Cij(u) adjacent to the points u and parallel to the coordinate planes B>ij. Thus, x has no stationary directions, the stationary directions of Vi are {i}, and the stationary directions of are {z, j } . A Goursat problem can be posed by prescribing the values of x at the point 0 E Z M, the values of Vi on the axes 23*, and the values of Cij,Cji on the coordinate planes 'Bij. The essential dimension of this discrete system is d = 3, independently of M . In particular, it may well be d < M . Consistency of this system for any M > 3 is interpreted as its integrability.

5.2. Approxim ation in discrete hyperbolic system s To handle approximation results for discrete geometric models, we need to introduce small parameters into hyperbolic systems of partial difference equations. The domain of our functions becomes *Be =

6 i Z X • • • X 6 A /Z .

If 6i = 0 for some index z, the respective component in is replaced by R. For instance, if c = (0 , . . . ,0), then 3 e = KM. Thus, the domains B>e possess continuous and discrete directions, with mesh sizes depending on the parameters e;. The definitions of translations and difference quotients are modified for functions on !Be in an obvious way: 1

5.2. Approximation in discrete hyperbolic systems

191

If €i = 0, then Si is naturally replaced by the partial derivative d{. For a multi-index a = ( a i , . . . , % ) , we set Sa = 6 *1 •••S ^ 1. The definition of elementary cells 6^, carrying the variables ified as follows:

is mod­

Cfc — | ^

Hiei : Hi £ [0, e*]| iGSfc (so that the cell size shrinks to zero in the directions with e* = 0). We see how the discreteness helps to organize the ideas: in the continuous case, when all Ei — 0, all the functions x k live at points, independently of the dimensions #S/e of their static spaces. In the discrete case, when all e* > 0, one can clearly distinguish between functions living on vertices (those without static directions), on edges (those with exactly one static direction), on elementary squares (those with exactly two static directions), etc.

Having in mind the limit e —> 0, we will only treat the case when the first m < M parameters go to zero in a uniform way, e\ = ••• = em = e, while the other M — m ones remain constant, em+i = •••= €m — 1- In this case 236 = (eZ)m x Z M_m, and we set 23 = 23° = Rm x Z M-m. Assuming that the functions = ge k •on the right-hand sides of (5.1) depend on e smoothly and have limits as e —> 0, we will study the convergence of solutions x e of the difference hyperbolic system (5.1) to the solutions x° of the limiting differential (or differential-difference) hyperbolic system (5.10)

dixk

=

gkti(x),

i G £*; n { 1 , . . . , m },

(5.11)

5iXk

=

gk>i(x),

i G £fc n {m + 1 , . . . , M } .

Naturally, (5.10), (5.11) describe the respective m-dimensional smooth ge­ ometry with M — m permutable transformations. Throughout this chapter, a smooth function g : D —> X is a function that is infinitely differentiable on its domain, g € C °°( 2)). For a compact set X C D, we say that a sequence of smooth functions ge converges to a smooth function g° with order 0(e) in C°°(3C) if \\d€ ~ 9°\\c£(%) ^ c?e

with suitable constants q for any £ G N. Convergence in C °°( 2)) means convergence in C°°(3C) for all compact sets X C D.

Definition 5.6. (Approximation of hyperbolic equations) We say that the difference hyperbolic system (5.1) approximates the differential (or differential-difference) hyperbolic system (5.10), (5.11) if ge ki —> gki with order 0(e) m C °°(X /c).

Convergence of discrete functions (defined on lattices 23e with different e) is understood as follows. We say that a family of discrete functions

5. Approximation

192

x e : 23e —> X converges to a function x° : 23 —> X with order 0(e) in C°°(23)

if for any multi-index a — ( a i , . . . ,

oim ),

sup 15a(xe — £°)(u)| <

Ca 6

with some constants ca. The symbol x° on the left-hand side is understood as a restriction of x to 23e C 23. Finally, we mention that we are mainly concerned with local problems, so we actually work with bounded domains of the lattices 23e, 23e(r) = {u G 23e : Ui G [0, r] for 1 < i < m, U{ G {0 ,1 } for m + 1 < z < M } . Each 23e(r) only contains finitely many points (though their number grows infinitely as e —» 0 if r remains fixed). For functions x e which are only defined on a bounded lattice domain 23e(r), the notion of convergence is modified in an obvious way: the supremum is taken over those lattice sites u where the respective difference quotient Sax e(u) exists. Now the fundamental statement about convergence of solutions of dis­ crete hyperbolic systems to solutions of the hyperbolic systems of partial differential equations can be formulated.

Theorem 5.7. (Convergence of solutions for discrete hyperbolic equations) Consider a Goursat problem (5.1), (5.2) for a hyperbolic system of difference equations. Suppose that:

i) discrete system (5.1) is consistent for all e > 0; ii) discrete system (5.1) approximates the differential (-difference) sys­ tem (5.10), (5.11) with order 0(e) in C °°(X k ); iii) Goursat data X^ converge to smooth functions X JP with order 0(e) in C°°(2s(k)(r)). Then there exists f G (0, r] such that, for e > 0 small enough, solution x e k of the Goursat problem exists and is unique on ® e(f); moreover, solutions xe k converge to smooth functions x® with order 0(e) in C 00(23(f)); these functions x® serve as a unique solution on 23(f) of the Goursat problem for

(5.10), (5.11) with the Goursat data X j?. In condition ii), convergence gek •—►gki in C°°(Xk), i-e-> on every com­ pact subset of Xfc, is assumed for simplicity of presentation only. In applica­ tions, functions geki are often defined on certain subdomains V ek C X/c, with the property that is open and dense in X&. In such a case, one requires in ii) the convergence gk i -+ g{k i 'm C°°(2)0). Then conclusions of Theorem 5.7 hold for generic initial data.

5.2. Approximation in discrete hyperbolic systems

193

As for condition iii), smooth data : 23gfc —> X/~ are usually given a priori, and discrete data X k are obtained by restriction to the lattice: Xk= . In such a situation, condition iii) is fulfilled automatically. We will not present a complete proof of Theorem 5.7 here but rather a substantial part of it illustrating all necessary technical ideas. Namely, we will provide the arguments for the simplest situation M — m — 2, and in this particular case we will only demonstrate the convergence x ek — x® —» 0 for solutions themselves and omit the proof of Sa(xek —x k) —> 0 for multi-indices a with \a\ > 1. In other words, we will prove the C°(23)-convergence instead of C°°(23)-convergence. The proof of the C°°(!B)-convergence for general M requires technical care but no essentially new ideas. Thus, we consider the discrete hyperbolic system (5.12)

S2a = / e(a,6),

with smooth functions (5.13)

a(ui, 0) = A e(ui ),

<Si& = ge(a, 6), and with the Goursat data 6(0, u2) = B t(u2)

for Ui G 23l(r). It is supposed that the functions / e, ge satisfy (5.14)

/ e(a, b) = f°(a , b) + 0(e),

ge(a, b) = g°(a , b) + 0(e),

uniformly 011 compact subsets of X x X, and that relation analogous to (5.14) holds for all partial derivatives of the functions / e, ge. Further, it is assumed that the discrete Goursat data (5.13) also have smooth limits: (5.15)

^ £(ui) = A °(«i) + 0(c),

B t(u2) = B °{ u2) + 0(e),

uniformly for Ui G 23\{r). Then the solutions (a€, be) of the Goursat problems for system (5.12) converge uniformly in 23(f), with a suitable f G (0, r], to a pair of Lipschitz functions (a0, 6°), (5.16)

ae(u i,u 2) = a°(u i,^ 2) + 0(e),

b€(u\,u 2) = b°(ui,u 2 ) + 0(e),

which constitute the unique solution in 23(f) of the Goursat problem for the system (5.17)

d2a = f°(a ,b ),

d\b = g°(a,b),

with the Goursat data (5.18)

a(Ul,0) = A °(Ul),

b(0 ,u 2) = B °(u 2)

for Ui £ 3 i(r).

Lemma 5.8. (A priori estimate) Let the norms of Goursat data A e, B e be bounded by e-independent constants.

Then there exists some f G (0, r]

such that the norms of the solutions (ae,6e) of the Goursat problem (5.12), (5.13) are bounded on 23e(f) independently of e.

194

5. Approximation

Proof. Let \A(-1, |J5e|< Mo, and choose Mi > Mo arbitrarily. Define f = (Mi - M 0)/sup c:

sup

{\fe(a,b)\ + \ g c(a,b)\}.

ja 1, |6] < A/1

We show that |«e|, \bf \ < M\ on ¥>' (f). Rewrite (5.12) as (5. 19)

ac(wi,U2 + e)

=

a( (111,112) + e f e(ac(u\,ii2),be(ui,ii2)),

(5.20)

6e(ui + e , u 2)

=

b( (in,U2) + tge(ac(u\,U2),b( (111,112)),

and then conclude by induction that \ae(u\,u2)\

<

|6£(«i,«2)| <

M() + (Mi - M0) — < Mx, r

A/o4 (M,

M {l) " ] < M,. r

for ( u i , v , 2 ) G ® e(r).

□

Remark. In general, one cannot expect that r = r because the solutions of the continuous Goursat problem may develop blow-ups that are absent in the discretization. However, if /% ge possess a global Lipschitz constant, then, as for ordinary differential equations, no blow-ups are possible, and one can take f — r in Lemma 5.8 and also in Theorem 5.7. Lemma 5.9. (Discrete Gronwall lemma) Assume that a nonnegative function A : Z + —>R satisfies

(5.21)

A(n + l) < (1 + K )A (n ) + /c,

with nonnegative constants K and k for all n > 0. estimate holds:

(5.22)

Then the following

A(n) < (A(0) + nn) exp(Kn).

Proof. This is proven by an easy induction, using the inequality 1 + K < exp (K ).

□

Lemma 5.10. (Lipschitz estimate) Assume that the initial data A0, B ° of the continuous Goursat problem are C 1 functions, and that the initial data A e, B € of the discrete Goursat problem satisfy (5.23)

\Ae(ii[) - j4°(ai)|< Me,

|Be(7x2) - £°(u2)| < M e,

with an e-independent constant M . Then the difference quotients S\a(\ d20 (\ S\b\ S2be of the solution of the discrete Goursat problem are bounded on 3 6(f) uni­ formly in 6, where f G (0,r] is chosen according to Lemma 5.8.

5.2. Approximation in discrete hyperbolic systems

195

Proof. By virtue of (5.12) and Lemma 5.8 it is clear that the difference quotients 62a€ and 5\be are uniformly bounded. Let the solutions (ae,b€) of the discrete Goursat problems be bounded by M i, and set M 2 = sup £

sup

||/e(a,6)|, \ge(a,b)\, |5a/ £(a,6)|,

\dbge{a,b)\],

|a|,|6|
}

which is finite since / e —> f ° and g€ —> g° locally uniformly in C l . Without loss of generality, M > M\ and M > M 2. We now prove the estimate for 5iae. Proceeding from u2 to u2 + e, we find: \6 iae(u i,u 2 + e)\

<

|<5iae(ui, u2)| + e\6 i f e(at(u i,u 2),tf(u i,u 2))\

<

|^ia£( « i ,1/2)1 + tM(\Siae(u i,u 2)\ + |<$i&£(ui,U2)|)

<

(1 + eM)|<5iae(ui, u2)| + eM 2.

Now Lemma 5.9 yields for all \u2\ < f: \Siae(u i,u 2)\ < (\5iAe(ui)\ + M 2f) exp (M r).

Under the assumptions of the lemma, we find: | < M £(ui)| < M V i ) |

+ ( I ^ ( m + e) - A ° ( m + e)| + ^ ( m ) - A ° ( u i ) | ) /e

< 3M , which yields the desired estimate: \5iae(u i,u 2)\ < (3M + M 2f) exp(Aff).

The same reasoning applies to 52b€.

□

Proof of Theorem 5.7. We give this proof for M = m = 2 only. We consider the sequence {(ae, be) } e=2-k with members extended to functions ®(f) —> X x X by linear interpolation. By Lemma 5.10, there is a constant L > 0 such that |ae(i4,U2) ~

(u i , u2)| + |^ ( ^ , ^ 2) - be(u i,u 2)\ < L(\u\ - m\ + \u2 - u2\).

In combination with Lemma 5.8, it follows that the family is equicontinuous and therefore satisfies the hypothesis of the Arzela-Ascoli theorem. Consequently, there exists a subsequence {(ae, be) } e=2-kn which uniformly converges to continuous functions a0, b° : 2J(r) —> X. Moreover, (a0, 6°) sat­ isfy the above Lipschitz condition with the constant L. To show that (a0, b°) solve the differential equations (5.17), observe that relation (5.19) and the Lipschitz property of ae imply [ u2/ e] - l

(5.24)

ae(ui, u2) = A e(u\) + e

^ k=0

f e(ae(ui,ke), b€(u\,ke)) + 0(e)

5. Approximation

196

for (u\,u2) E 2 (f). As the convergence of the subsequence {(ae, be) } e=2-kn is uniform, and f e —►/ in C 1, one can pass to the limit e —> 0 on both sides of (5.24) to obtain nU2

(5.25)

a °(u i,u 2) = i4°(ui) +

f ° (a°(uu r}),b°(ui,ri))dr}.

Jo It follows that a0 is differentiable with respect to u2, with d2o0 = /°(a °, 6°). The function 6° is treated in the same manner. Now, estimate (5.16) can be proven for an arbitrary e — 2 ~k. Define the approximation error A e(n) — max{\ae(u i,u 2) - a °(u i,u 2)\ + \be(u i,u 2) - b°(u i,u2)\ : { u i , u 2) E ® e(f), u\ + u2 = ne}. Combining formula (5.19) with integral representation (5.25) yields A e(n + 1)

<

A e(n) + \Ae - A°\{ne + e) + \Be - B°\(ne + e) +6

<

max {\Si(ae - a°)\(ui,u2) + |^2(6e - b°)\(ui,u2)\ ui+U2=ne v

A e(n) + e

max

Ul+U2=n€

{| /€(ae,6£)(ui,n2) - /° ( a ° , 6°)(«i, u2)|

+\ge(at,bt)(u i,u 2) - g°(a 0 ,b 0 )(u 1,u 2)\} + 0(e)

<

(1 + 0(e))A£(n) + 0(e).

By Lemma 5.9, we obtain A e(n) = 0(e) for ne < f. This implies the estimate (5.16). □

5.3. Convergence of Q -nets Discretization of a conjugate net. Recall that a conjugate net / : Mm —> WN can be determined by the data (Qi,2) (see Section 1 . 1), while a Q-net f e : (eZ)m —► can be determined by the initial data (Q^2) (see Section 2 . 1). We now demonstrate how to produce from the smooth data (Qi,2) certain discrete data (Q f2)> which will assure the convergence of the corresponding Q-nets to a given smooth conjugate net. Define the discrete curves f € \&. by restricting the curves f\ ^ . to the lattice points: f e(u) = f(u ),

u E ®|,

Similarly, define the plaquette functions

1 < i < m. by restricting

lattice points: c€ i:j(u) = Cij (u ),

uE

1 < i ^ j < m.

to the

197

5.4. Convergence of discrete Moutard nets

An even better option is to read off the values of the corresponding plaquettes of 23^-: c\j(u) = Cij(u + \ei + fej),

u e 23^,

at the midpoints of

1 < i^= j < m.

Either choice gives initial data (Q^2) which define an 6-dependent family of Q-nets f e : (eZ)m —> RN , called canonical Q-nets corresponding to the initial data (Qi,2).

Theorem 5.11. (Convergence of Q-nets) There exists r > 0 such that the canonical Q-nets f e : 23e(r) —►RN converge, as e

0, to the unique

conjugate net f : 23(r) —> RN with the initial data (Qi,2)- Convergence is with order 0(e) in C °°(®(r)).

Proof. This follows directly from Theorem 5.7, since systems (2.1), (2.7) and (1.1), (1.2) are manifestly hyperbolic (and can be easily rewritten in the first order form).

□

Discretization of an F-transformation. Recall that an F-transform of a given conjugate net is determined by the initial data (Fij2) (see Section 1.1). We now produce from these the initial data (F f2) (see Section 2.1) for an e-dependent family of F-transforms of canonical Q-nets corresponding to the initial data (Q i ,2)Take the point /+ (0 ) from (Fi). Define the edge functions a\\w, by restricting the functions a* f^., bi to the lattice points, or, better, to the midpoints of the corresponding edges of 23-. This gives the data set (F f2); along with the data (Q f2) produced above this yields in a canonical way an e-dependent family of Q-nets F e : (eZ)m x {0 ,1 } —» R N, which will be called the canonical Q-nets for the initial data (Qi,2), (F\2).

Theorem 5.12. (Convergence of discrete F-transformations) The canonical Q-nets (f e)+ = F €(-,l) : Be(r) —►RN converge to the net : 23(r) —> R n which is the unique F-transform of f with the initial data (Fij2). Convergence is with order 0(e) in C°°(23(r)).

Proof. Again, this follows directly from Theorem 5.7 applied to the hyper­ bolic systems consisting of (2.10)-(2.12) in the discrete case and of (1.12)— (1.14) in the smooth case. Note that the discrete equations are implicit, and their solvability for e small enough is guaranteed on the subset of the phase space, {aj ^ 0 : 1 < j < m}, which is open and dense. □

5.4. Convergence of discrete M outard nets Discretization of a Moutard net. Given a Moutard net y : M2 —> R N defined by the initial data (Mi 2) (see Section 1.2.1), we produce initial data

5. Approximation

198

(M^2) for an 6-dependent family of discrete M-nets y e : (cZ)2 —> R N (see Section 2.3.10). Discrete curves y c\'%« are obtained from the smooth curves y\% by restricting to the lattice points: y e{ u ) = y ( u ) ,

u e 'B t ,

i = 1, 2 .

The plaquette function a\2 : (eZ)2 —> R is obtained from the function q\2 restricted to the lattice points: a\2 (u) = 1 + \e2qi2 (u),

u € (eZ)2

(one could also restrict qu to midpoints of the corresponding plaquettes). Now canonical discrete M-nets y e : (eZ)2 —> RN for a given Moutard net y are defined as the solutions of the difference equation (2.56) with the above data (M^2)*

Theorem 5.13. (Convergence of discrete Moutard nets) Canonical discrete M-nets y c : 23e(r) —> WLN converge, as e —> 0, to the unique M-net y : ®(r) —►R N with the initial data (Mi^). Convergence is with order 0(c) m C ° ° (®(r)).

’

Proof. Equation (2.56) is manifestly hyperbolic (and can be easily put in the first order form). It approximates equation (1.29), because it can be rewritten as

tifay = 5 <712(ny + T2y) = qi2{y + f (Siy + § s 2y). Now Theorem 5.7 can be applied.

□

Discretization of a Moutard transformation.

Let the initial data (M Ti>2) for a Moutard transformation be given (see Section 1.2.2). De­ fine the edge variables b\\^. from the functions P il^ restricted to the lattice points: 6f(ui,0) = 1 + cpi(t4i , 0 ),

62(0 ^ 2) = 1 + ^ 2 (0 , u2),

Ui G cZ

(one could restrict pi \^. to the midpoints of the corresponding edges, as well). This gives us the initial data (M T f2) for canonical discrete M-nets {ye)+ : (eZ)2 —> R N (see Section 2.3.9).

Theorem 5.14. (Convergence of discrete Moutard transformations) Canonical discrete M-nets (ye)+ : 23e(r) —> RN converge to the unique Moutard transform y+ : !B(r) —» R N of y with the initial data (M Ti^). Convergence is with order 0(c) in C 00^ ^ ) ) .

Proof. The system consisting of (2.58), (2.59) is hyperbolic. Upon substi­ tuting bi = 1 + epi and a\2 = 1 + \e1q\2 > these equations can be rewritten as 5iy+ + h y = Pi(y+ - n y ) ,

S2y + - S2y = p2 {y+ + r2y),

5.5. Convergence of discrete asymptotic nets

199

and 1 + er2pi _ 1 + €Tip2 _ 1 + (c‘2/2) q^2 ______________ 1____________ 1 + epi

l + e p -2

l + (f2/2)gi2

1 + e2(ql2 ~ P1P2 ) + 0(e3) '

Clearly, they approximate, as e —> 0, equations (1.30).(1.31) and (1.32)(1.33), respectively. It remains to apply Theorem 5.7. □

5.5. Convergence of discrete asym ptotic nets Discretization of an A-surface. Initial data ( A ^ ) for an A-surface (see Section 1.3) are nothing but initial data (Mi,2) for the Lelieuvre normal field n : R 2 —> R3. Thus, discretizing the latter as described in Theorem 5.13, we arrive at the canonical construction of the initial data (A^2)> which give a converging family of the discrete Lelieuvre normal fields ne : (eZ)2 —> R3 (see Section 2.4.2). Equations (2.69) define the discrete A-surfaces f e : (eZ)2 —► R3, called the canonical discrete A-surfaces corresponding to the initial data (Al,2>).

Theorem 5.15. (Convergence of discrete asymptotic nets) Canonical discrete A-surfaces f e : (B c(r) —> M3 converge, as e —> 0 , to the unique Asurface f : !B(r) —►M3 with the initial data (Ai,2). order 0 (e) in C °°(®(r)).

Convergence is with

Proof. Equations (2.69) are hyperbolic and they approximate equations (1.38). Theorem 5.7 can be applied to prove the convergence of / e, after the convergence of ne has been already proven. □

Discretization of a Weingarten pair. Initial data (W 12 ) for a Wein­ garten transformation (see Section 1.3) are nothing but initial data (MTi^) for a Moutard transformation of the Lelieuvre normal field. The construc­ tion of Theorem 5.14 delivers the initial data for a family of Lelieuvre normal fields (ne)+ : (eZ)2 —> M3, which are therefore seen as the data (W ^2) f°r transformed A-surfaces ( f e)+ : (el ) 2 —> M3, obtained via (2.70) (see Section 2.4.3).

Theorem 5.16. (Convergence of discrete Weingarten transforma­ tions) Canonical discrete A-nets ( / e)+ : Be(r) —» R3 converge to the unique Weingarten transform / + : ®(r) —►R3 of f with the initial data (W i^). Convergence is with order 0(e) in C°°(!B(r)).

Proof. This is proven by comparing the (identical) formulas (1.40) and (2.70), after the convergence of rf and (ne)+ has been established. □

200

5. Approximation

5.6. Convergence of circular nets In this section, we address the problem of approximating smooth orthogonal nets by discrete circular nets. Recall that the former are governed by the system (1.44)-(1.47) with constraint (1.48), while the latter are governed by similarly looking equations (3.12)-(3.14), (3.17) with constraint (3.18). We demonstrate that this analogy can be given a qualitative content, so that for a given orthogonal net one can construct an approximating family of circular nets. However, there is a substantial obstruction to accomplishing this, which can be seen by a careful comparison of the constraints (1.48) and (3.18). We think of smooth rotation coefficients as being approximated by discrete ones. But since the discrete rotation coefficients fikj only have i 7^ k ,j as evolution directions (that is, they are plaquette variables at­ tached to elementary squares parallel to ®j&), there is seemingly no chance to get an approximation of such smooth quantities as diftij involved in the smooth orthogonality constraint (1.48). In order to be able to achieve such an approximation, we need some discrete analogs of the smooth rotation coefficients which would live on edges. For this aim, we turn to the Mobius-geometric description of circular nets from Section 3.1.4, more precisely, to the frame equations (3.25). Introduce vectors Vi — 'ipVi'0 - 1 ; then the frame equations become (r^ ) ^ -1 = —e^V*. Expanding these vectors with respect to the basis vectors e^, we have a formula analogous to (1.53): (5.26)

Vi = i)Vii>~1 = (JiGi -

Pkie k + Mook^i

The fact that the e^-component here is equal to hi, is easily demonstrated. Indeed, from (3.25) it follows that Tif — f = hidi = hi(Tii/j)~lei'ip. Now equation (3.24) allows us to rewrite this equivalently as [eo, ( r ^ ) ^ -1 ] — /i*e?;, which proves the claim above. Observe also the normalization condition (5-27)

a2 = k^i

Coefficients pki are edge variables analogous to smooth rotation coefficients. Indeed, the vectors Vi are defined on the edges of Zm parallel to the coor­ dinate axis 3 i. However, these vectors do not immediately reflect the local geometry near these edges; rather, they are obtained by integration of the frame equations (3.25), and thus are of a nonlocal nature. Thus, in the discrete case we have two different analogs of the rotation coefficients: local plaquette variables (3ij for 1 < i / j < m, defined on the elementary squares of Zm parallel to and nonlocal edge variables pki for l < i < m , 1 < k < N , k ^ i, defined on the edges of Zm parallel to

5.6. Convergence of circular nets

201

Evolution equations for Vi are obtained from (3.13) and the frame equa­ tions (3.25): TiVj = Vjile i(V3 + PiJVi)e f

In the derivation one uses the identity vi(vj + (3jivi)vi = vj + PijVi, which easily follows from (3.18). The resulting evolution equations for the edge variables pkj read: (5.28)

npkj

=

vjil (Pkj + PkiPij),

(5.29)

Tipij

=

i/ji {~pij H- 2(Tifiij).

Here 1 < i ^ j < m, 1 < fc < iV, and k ^ i ,j . The circularity constraint (3.18) can be now written as (5.30)

fiij + (3ji = CTiPij + &jPji ~ ~ ^ ^ PkiPkj') k^i,j

and gives a relation between local plaquette variables faj and nonlocal edge variables pkj- The system consisting of (5.28), (5.29) and (5.30) can be regarded as the discrete Lame system ; cf. (1.47), (1.48). Turning to the problem of approximation, we start with approximation of a single orthogonal net / : Mm —> R N. For the approximating circular nets, we have M = m and all e* = e. In all the formulas of Section 3.1 and of the present section, we have to replace the lattice functions hi, fiij, pkj by ehi, eflij, ephj, respectively. Observe that formulas (3.16), (5.27) become Vji = Vij = (1 - e2f3ijf3ji )1/2 = 1 + 0(e2),

^ = ( 1" t E

Pki) 7 = 1 + ° ( e2)-

k^i

Under this rescaling, equations (3.12), (3.13), (3.14) and (3.17) can be put into the standard form (5.1) with functions on the right-hand sides approxi­ mating, as e —> 0 , the corresponding functions in equations (1.7)—(1.10) with order 0 (e). Nevertheless, Theorem 5.7 still cannot be applied to orthogonal nets. The reason for this is that the full system of differential equations describing orthogonal nets, consisting of equations (1.7)—(1.10) and constraint (1.54), is nonhyperbolic. Its nonhyperbolicity rests on the fact that the constraint (1.54) is not resolved with respect to the derivatives difiij. Note, how­ ever, that constraint (1.54) does not take part in the evolution of solutions starting with the data given in the coordinate planes the constraint is satisfied automatically, provided it is fulfilled for the coordinate surfaces / h v Therefore, we will obtain a convergence result for orthogonal nets as soon as it will be established for coordinate surfaces.

202

5. Approximation

Discretization of a curvature line parametrized surface. Initial data for a smooth curvature line parametrized surface / : 2312

are:

(i) two smooth curves / [3 . (i = 1, 2 ), intersecting orthogonally at /(0 ); (ii) a smooth function 712 : 2312 —> M, whose designated meaning is

712 =

^(d\(3i2 ~

^ 2/^21 ).

Let f\
/(0) = ^ -1 (0)e0V’(0),

0j(0) = V,_1(0)eji/,(0)

(i = 1,2).

Define the frames ^ : 23* —> 3-Coo of the curves f \<si as the solutions of equations (1.51) for i = 1,2 (considered as ordinary differential equations) with the initial value ^(0). Rotation coefficients of the curves /[ $ . are the functions fiki : 23; —> R defined by the formula (1.53) for i = 1,2. Define the discrete coordinate curves f e\w by restricting the functions f\'b- to the lattice points. Let h\ = \5if€\and v\ = (h\)~l 8i f c be the discrete metric coefficients and unit vectors along the discrete curves. Define the frame i/je : 23^ —> IK00 by iterating the difference equation (3.25) for i — 1,2 with initial condition i/je(0) — ijj(0). Then canonical rotation coefficients of the discrete curves f e\< Be. are the coefficients pki : 23^ —>M in the expansions K = W i i P ) - 1 = ^ie< - \

+ eK e ook^i

Finally, let the plaquette function : 23^2 —>K be obtained by restricting 712 to the lattice points (or to the midpoints of the corresponding plaquettes o f ® i 2).

Thus, we get valid Goursat data for a hyperbolic system of first order difference equations for the variables / e, h p e kv consisting of (3 . 12), (3.13), (3.14), (5.28) and (5.29) with distinct i, j G {1, 2} and 1 < k < TV, where the following expressions should be inserted: &12 — a \ P l 2 ~

2 (2

'S ^ J PklPk2~ k> 2

7i2) ’

021 — a 2 P 2 l ~ ^

(2

PklPk2

+ 712 ) -

k> 2

The nets / e : 23|2 —^ Qo^ defined as solutions of the Goursat problem just described are circular surfaces, since they fulfill the circularity constraint (5.30). They will be called canonical circular surfaces constructed from the above initial data.

Theorem 5.17. (Convergence of circular surfaces) There exists r > 0 such that the canonical circular surfaces f € : 23f2(r) —> Qq converge, with

5.6. Convergence of circular nets

203

order 0 (e) in C °° (l*>\2 (r)), to the unique curvature line parametrized surface f : ® i 2(r) —» Qgf with the initial data f\^. (i = 1,2) and ^(diPn —82 ^ 21) = 712. Edge rotation coefficients pki and plaquette rotation coefficients (3{2, /?2i of the circular surfaces f e converge to the corresponding rotation coefficients 0 ki of the curvature line parametrized surface f .

Proof. We begin with showing the convergence of the frames, of the rotation coefficients, pki —►0ki , along the discrete curves

and This

follows from two observations. First, t)f(0) ~ ^(0) + f (<9z^)(0) + 0 ( e2)> so that i n - 1 )# (0 ) =

ejV’(0)(5iUi)(0) + 0(e2).

Second, combining frame equations on two neighboring edges of !B|, one finds that (Ti -

everywhere on theory.

T

~

=

- e ^ i

1 - Tj- 1 )#- =

-e e i-ip ^ d iV i)

+ 0(e2)

The claim follows by standard methods of the ODE

Now an application of Theorem 5.7 shows that the functions f e :

—*

Q q converge to the functions / : $ 12

which solve the Goursat prob­ lem for the hyperbolic system of first order differential equations, consisting of (1.44)-(1.47) with distinct z, j G {1 ,2 } and 1 < k < AT, and diP\2 =

^ 2 & k iP k 2 + 712,

k> 2

cfe/fei = “ 2

” ^ 12'

k> 2

Solutions f3ki satisfy the relation \(d\ft\2 — ^2^ 21) = 712 and the orthogo­ nality constraint (1.48). □

Discretization of an m-dimensional orthogonal net. Given the initial data (Oi?2) for an m-dimensional orthogonal net (see Section 1.4), we can apply the procedure described in the previous paragraph, with an initial frame ^(0) G 3-Coo such that /(o ) = -0_1(O)eo^(O),

V i(0 )

= ip~l ( 0 ) ^ ( 0 )

(1 < i < m),

to produce, in a canonical way, the circular surfaces / er^

and their pla-

quette rotation coefficients /?^. Thus, we get the data (Of^2) (see Section 3.1) for an e-dependent family of circular nets f e : (eZ)m —> Q q . These nets will be called the canonical circular nets corresponding to the initial data (Ol,2).

Theorem 5.18. (Convergence of circular nets) The canonical circular nets f € : ® e(r) —> R N converge, as e —►0 , to the unique orthogonal net

204

5. Approximation

f : ®(r) —> R n with the initial data (Oi^)- Convergence is with order 0(e)

m C °°(B (r)).

’

Proof. The data (Ofj2) yield a well-posed Goursat problem for the hyper­ bolic system of first order difference equations for the variables / e, £f, h\, Pfj, consisting of (3.12), (3.13), (3.14), (3.17). The convergence of these Goursat data is assured by Theorem 5.17. Now the claim of the theorem follows directly from Theorem 5.7. □

Discretization of a Ribaucour transformation. Given the initial data (Ri,2) for a Ribaucour transform of an orthogonal net (see Section 1.4), de­ fine the plaquette rotation coefficients (3eMi on the “vertical” plaquettes along the edges of the coordinate axes by restricting the corresponding func­ tions cOi to lattice points or, alternatively, to midpoints of the corresponding edges of (3€ Mi(u) = eOi(u) or e0i(u + §),

w G ?•,

1 < i < m.

Thus, we get the data (R^2) (see Section 3.1), which, together with (Of^2), allow us to construct in a canonical way circular nets F e : (eZ)m x {0, 1} —> . They will be called the canonical circular nets corresponding to the initial data (Oi^), (Ri,2)-

Theorem 5.19. (Convergence of discrete Ribaucour transforma­ tions) The canonical circular nets ( / e)+ = F e(•, 1) : ® e(r) —> R N converge to the unique Ribaucour transform / + : ®(r) —> R N of f with the initial data (Ri,2)- Convergence is with order 0(e) in C°°(!B(r)).

Proof. Define veM {0) as the unit vector parallel to S f( 0) = / + ( 0) — /(0 ), and set heM (0) = \Sf(0)\. These data along with Pc Mi on the coordinate axes, added to the previously found f e(0), v\, h\, for 1 < i, j < m, form valid Goursat data for the system (3.12), (3.13), (3.14), (3.17). The circularity constraint (3.18) implies that /?zeM = —2(v b v%i) — eOi on all edges of *B■. Perform the substitution VM —

y + 0 ( e)>

h eM = £ + Q ( e ) ,

f5e Mi

= e Q i + 0 ( e 2),

— —2 ( v i , y ) + Q ( e )

in equations (3.13), (3.14), (3.17) with one of the indices equal to M . Taking into account that in this limit one has UiM = VMi = 1 -

e(Vi> V )° i + ° ( e2)>

one sees that the limiting equations coincide with (1.57), (1.58), (1.59). A reference to Theorem 5.7 finishes the proof. □

205

5.7. Convergence of discrete K-surfaces

5.7. Convergence of discrete K-surfaces Discretization of a K-surface. Given the initial data (K) for a K-surface (see Section 1.6), we define the initial data (KA) (see Section 4.2) for an edependent family of discrete K-surfaces with t\ — e2 — e by restricting n\^{ to the lattice points, as for general A-surfaces. Thus, we get two intersecting discrete curves in § 2. Define discrete M-nets ne : (eZ)2 —►§ 2 as solutions of the difference equations (4.31) with the initial data (KA). Finally, define the discrete K-surfaces f e : (eZ)2 —> R3 with the help of the discrete Lelievre representation (2.69). These will be called the canonical discrete K-surfaces corresponding to the initial data (K).

Theorem 5.20. (Convergence of discrete K-surfaces) Canonical dis­ crete K-surfaces f e: 23e(r) —> R3 converge, as e —> 0, to the unique K-surface / : 23(r) —►R3 with the initial data (K). Convergence is with order 0(e) in C°°0B(r)).

Proof. We have for n — ne: e _ (n, Tin + t 2u) _

12

1 + (nn, t 2u)

2 + e(n, 8 \n + S2n) 2 + c(n, <5in + 52n) + e2 (Sin, ^2^) *

Since (n, ^n) = 0(e), we find that af2 = 1 — ^€2(5in, 52n) + 0(e4). Comparing this with (1.68), we see that Theorem 5.7 can be applied and yields convergence of the net ne: (eZ)2 —> § 2 to the smooth net n: M2 ^ S2. Finally, convergence of f e to / follows exactly as for general A-surfaces. □

Discretization of a Backlund pair. Let the initial data (B) for a Backlund transformation of a given K-surface / , i.e., the point n+ (0), be given. Take it as the initial data (BA) for the discrete Backlund transformations ( / e)+ : (eZ)2 —> M3 of the family f e of discrete K-surfaces constructed in Theorem 5.20.

Theorem 5.21. (Convergence of discrete Backlund transformations) Canonical discrete K-surfaces ( / e)+ : ® €(r) —> M3 converge to the unique Backlund transform f + : !B(r) —> R3 of the K-surface f with initial data (B). Convergence is with order 0(e) in C°°(!B(r)).

Proof. For the Backlund transformation, equations (4.43), (4.44) hold. In the smooth limit we find: ^ «

{Sin, n+ + n)

(Sin,n+ )

1 — (nn, n+)

1 — (n,n+)

(<52n ,n + - n ) = - 1 + (T2„ , „ +)

=

6’

(<52n ,n + ) - r r K ^ ) +0<£)'

5. Approximation

206

Comparing this with (1.41)-(1.42) and applying Theorem 5.7, we prove con­ vergence of the Gauss maps. □

5.8. Exercises 5.1. Check that each of the following four difference equations approximates the ordinary differential equation dx = f(x ) for x : E —> X (where d stands for the ordinary derivative d /d t ):

for x : eZ —> X, where Sx(t) = (x(t + e) — x(t))/e. Use the implicit function theorem where appropriate.

5.2. Put the Hirota equation for E, e2

1

sin - ( t i t 2 -

T\(j> -

r 2(j) + (j>) — —

1

sin - ( r i T 2 0

+

+ r 20 + 0 ) ,

and the sine-Gordon equation for (f) : M2 —>E, ^ 1^20 = sin 0 , into the form of hyperbolic first order systems, and check that the former approximates the latter as e —> 0 .

5.3. Put the difference equation for discrete Lorentz-harmonic functions n : (eZ)2 —> § 2,

(n,rin + r2n) Tir2n + n = ------ -------------r (Tin + r2n), l -h( r in ,r 2n) and the differential equation for Lorentz-harmonic functions n : E 2 —> S2, 9i92n = —(din, d2n) n, into the form of hyperbolic first order systems, and check that the former approximates the latter as e —> 0. Why would the approximation claim fail for similar equations in the case of functions with values in L ^ 1,1?

5.4. Prove the following form of the discrete Gronwall lemma: Let A, a, b : Z+ —> E+ be three nonnegative sequences satisfying n—1

A(n) < an + ^ 2 h&(k)k=l

Then

n—1

A(n) < an +

n—1 ak h

k=0

n (H j=k+l

An interesting (and important) particular case is that of constant coefficients an = k and bn = K .

5.9. Bibliographical notes

207

5.9. Bibliographical notes Geometric convergence theorems are available in the literature for problems described by elliptic partial differential equations, such as the Plateau prob­ lem in the theory of minimal surfaces; see, for example, Dziuk-Hutchinson (1999). Convergence of metric and geometric properties of general polyhe­ dral surfaces was shown in Hildebrandt-Polthier-Wardetzky (2006) based on the analysis of the “cotan” Laplace operator. For surfaces described by hyperbolic partial differential equations, first approximation results were obtained in Bobenko-Matthes-Suris (2003, 2005). The presentation of this section follows these papers. The complete proof of the main approximation Theorem 5.7 can be found in Matthes (2004). A related purely geometric construction of circular nets approximat­ ing general curvature line parametrized surfaces is given in Bobenko-Tsarev (2007).

Chapter 6

Consistency as Integrability

Up to now we have encountered many instances of multidimensional nets which serve as discretizations of smooth geometries traditionally associated with, and described by integrable systems. The idea of consistency (or com­ patibility) is in the core of the integrable systems theory. One is faced with it already at the very definition of the complete integrability of a Hamiltonian flow in the Liouville-Arnold sense, which means exactly that the flow may be included into a complete family of commuting (compatible) Hamiltonian flows. It is impossible to list all applications or reincarnations of this idea. We mention only some of them relevant for our present account.

• In the theory of solitons nonlinear integrable equations are repre­ sented as a compatibility condition of a linear system called the zero curvature representation (also known as Lax, or ZakharovShabat representations). Various analytic methods of investigation of soliton equations (such as the inverse scattering method, algebrogeometric integration, asymptotic analysis, etc.) are based on this representation. • It is a characteristic feature of soliton (integrable) partial differ­ ential equations that they appear not separately but are always organized in hierarchies of commuting (compatible) flows. • Another indispensable feature of integrable systems is that they possess Backlund-Darboux transformations. These special transfor­ mations are often used to generate new solutions from the known ones.

209

6. Consistency as Integrability

210

In fact all these properties are interrelated and it is customary to understand the integrability as the presence of one (or some combination) of these fea­ tures. In this chapter we show how the development of discrete differential geometry leads to a new understanding of the very notion of integrability and its properties.

6.1. Continuous integrable systems Consider one of the most celebrated integrable systems having numerous applications in differential geometry as well as in mathematical physics, the sine-Gordon equation (6.1)

d\d2(f) = sin <6

for a function : E2 ^ E. Recall the geometric interpretation of the sineGordon equation. Let / : E2 —> E3 be a surface parametrized along its asymptotic lines. Surfaces of constant negative Gaussian curvature K = —1 (K-surfaces, for short) in the asymptotic lines parametrization are charac­ terized by the additional requirement that \d[f\ does not depend on u2 , and \d2f\ does not depend on u\. Reparametrizing the asymptotic lines of a K-surface if necessary, one can assume that \d\f\ = \d2f\ = 1. Then the angle (j) = cj)(u) between the vectors d\f and d2f satisfies the sine-Gordon equation (6.1). Integrability of the sine-Gordon equation has many manifestations, two of which will be of special importance for us: the zero curvature represen­ tation and the existence of Backlund transformations. To formulate the zero curvature representation of the sine-Gordon equa­ tion, consider the matrices

<6-2>

" = K -t

(6.3)

V'

=

-

t

) ■

1 ( A_ , %

They depend on u G E2 through the function (j) and its partial derivatives, and also depend on a (real) parameter A, known in the theory of integrable systems as the spectral parameter. It is usual to think about U, V as functions of u G E2 which take values in the twisted loop algebra 5 [A] = {£ : R*

su(2) : f ( - A ) = a3£(A)<73},

a3 = ( J

^

)

6.1. Continuous integrable systems

211

Then it is a matter of a straightforward computation to check that 0 is a solution of equation (6.1) if and only if the zero curvature condition (6.4)

d2U - d 1V + [U,V] = 0

is satisfied identically in A. The name “zero curvature” comes from the fact that (6.4) expresses the flatness of the connection (or, better, the oneparameter family of connections) on R2 given by the differential one-form Udui + Vdu 2 > This condition assures the solvability of the following system of linear differential equations: (6.5) for a function

diV = U 9 ,

d2
: R2 —> G[A] with values in the twisted loop group G[X] = { £ : R* —> •SU(2) : S ( - A ) = a3S(A)<73}.

The existence of the zero curvature representation is considered as one of the main integrability features of the sine-Gordon equation (and the likes). On a general note, it relates a nonlinear equation (6.1) to the system of linear equations (6.5), which are amenable to analysis. In particular, the spectral theory of the first equation in (6.5) lies in the basis of the inverse spectral transformation approach to the solution of certain boundary value problems for the sine-Gordon equation. Also conserved quantities (integrals) of the sine-Gordon equation can be derived directly from its zero curvature representation. Furthermore, the zero curvature representation allows one to reconstruct a K-surface corresponding to a solution 0 of the sine-Gordon equation. Given a solution 0 : R2 —> R, introduce the matrices (6.2), (6.3) satisfying (6.4). Define the function ^ : R2 —> G[A] as the solution of equations (6.5) with the initial condition ^ (0,0; A) = 1. Then the immersion / : R2 —> R3 obtained by the Sym formula,

(6 .6 ) under the canonical identification (4.9) of su(2) with R3, is an asymptotic lines parametrized K-surface, with the angle 0 between the asymptotic di­ rections. The function ^ is known as the extended frame of / . Moreover, the right-hand side of (6.6) with various A not necessarily equal to 1 delivers a whole family of immersions f\ : R2 —> R3, all of which turn out to be asymptotic lines parametrized K-surfaces. These surfaces f\ constitute the so-called associated family of / . The classical Backlund transformation is the next common feature of all known integrable systems. In the case of the sine-Gordon equation, it is given by the following construction. For a given solution 0 of (6.1), a

212

6. Consistency as Integrability

new solution
d\

+ d\4> = — sin a

2

— ,

d2(fi+ — d2(t> — 2a sin ^

^. 2

This system is compatible, ^ ( ^ i < /> + ) = <9i(<92+ ) , provided 0 is a solution of the sine-Gordon equation, and then M3 characterized, according to Definition 1.26, as follows: the straight line segments [ f ( u ) ,f + (u)\ are tangent to both surfaces / and / + , and their length is independent of u. It can be checked by a direct computation that equations (6.7) are equiv­ alent to the following matrix differential equations, which are satisfied iden­ tically with respect to the spectral parameter A: (6.8)

ft W - J7+W - W U,

<92W = V + W - WV,

where the matrix W is given by the formula ,# 0 ,

w _

( e '(* + - « /2

^

-iaX

-iaX ■

^

j'

On the other hand, (6.8) constitute a solvability condition for the system consisting of (6.5) and similar equations for the matrix function (6.10)

- W ®.

One can show that formed surface / + .

serves as the extended frame of the Backlund trans­

A remarkable property of Backlund transformations is given by Bianchi ’s is a Backlund transformation of (fi with pa­ rameter a and
(6.11)

sin \ (<^12^ + 4

— (fi^ — (fi) = — sin \ (<^12^ — 4^ a

4

+ (fi^ — (fi).

So, integrable systems, for which the sine-Gordon equation is a proto­ typical example, are characterized by such features as zero curvature repre­ sentation and Backlund transformations with permutability properties. The origin and the very existence of these features is considered in the classical theory of integrable systems as something mysterious and transcendental.

6.2. Discrete integrable systems

213

6.2. Discrete integrable systems The theory of discrete integrable systems has been developed for some time as part of the general theory of integrable systems. Its aims at the early stages were not very ambitious: just to find difference analogs of integrable differential systems, enjoying the same integrability features. In this in­ troductory section we give an illustration by the example of the integrable discretization of the sine-Gordon equation, known as the Hirota equation: 1 ( 6 .1 2 )

s in -(r iT

2(/> -

ri -

T20 +

0)

e2

=

1

— s i n - ( t i 720 + r\(f> +

t 2 +

0 ).

Here 0 is a real-valued function on (eZ)2, and the shift symbols stand for Tk <j)(u) = cf)(u + ee k). The Hirota equation (6.12) turns out to describe discrete K-surfaces, i.e., discrete A-surfaces / : (eZ)2 —►M3 with all edges of the same length ei, so that |<Si/| = \62f\ = L Here, of course, 5kf(u ) = (f(u + eek) - f(u ))/e . The discrete zero curvature representation of equation (6.12) is formu­ lated in terms of the matrices It, V : (eZ)2 —> G[A], defined by the formulas / (6.13)

IX =

e i(Ti-)/2

ie A

\

2~

1 *i(A)

ie A

D-i(Ti<j)-)/2

2~ ie d*(T2^+0)/2

2A (6.14)

V

-

e 2( x) \ 2A

/

where the normalizing factors ^i(A) = (1 + e2A2/4 ) 1//2 and ^(A) = (1 + 62A” 2/4 ) 1//2 are introduced in order to assure that U,V G G[A]. The matrix equation (6.15)

(r2U)V - (n V )U

is satisfied identically in A if and only if the function 0 solves (6.12). Equa­ tion (6.15) is called a discrete zero curvature representation of the Hirota equation (6.12). It expresses the flatness of a discrete G[A]-valued connec­ tion, given by the matrices IX assigned to the directed edges (u,u + ee\) and the matrices V assigned to the directed edges (u,u + ee2) of the lattice (eZ)2; see Figure 6.1. In its turn, this condition assures the solvability of the following system of linear difference equations: (6.16)

n tf = Utf,

for a function ^ : (eZ)2 —> G[A].

r2^ =

214

6. Consistency as Integrability

T2

r 2U

V

(f>

TiT2

T\V

U

Tl<(>

Figure 6 .1 . Discrete flat connection.

As in the continuous case, the discrete zero curvature representation can be used as a starting point for application of the analytical machinery of the inverse spectral methods. It also yields the conserved quantities (integrals) of the Hirota equation. Moreover, it can be used to reconstruct the underly­ ing discrete K-surface, corresponding to a given solution (fi : (eZ)2 —> R of the Hirota equation, in literally the same fashion as in the smooth case. Given a solution (fi of equation (6.12), introduce matrices (6.13), (6.14) satisfying (6.15). Define the function \I> : (eZ)2 —> G[A] as the solution of (6.16) with the initial condition ^ (0,0; A) = 1. Then the Sym formula (6.6) determines a net / : (eZ)2 —> M3, which is a discrete K-surface with the characteristic angle function (fi and with the edge length e£, where £ = (l + e2/4 ) - 1 . Again, the right-hand side of (6.6) for various A not necessarily equal to 1 delivers an associated family f\ of discrete K-surfaces. The Backlund transformation for equation (6.12) is given by the follow­ ing difference analogs of formulas (6.7): (6.17)

sin ^ ( ti 4>+ —
t\4> — 0)

=

(6.18)

sin ^ (t 20 + - (p+ - T2(p + )

=

s in ^ (n 0 + + 0 + - n 0 - 0),

^ sin ^ (t 2) ■

Statements analogous to those for the sine-Gordon equation hold. Differ­ ence equations (6.17), (6.18) are compatible; that is, ti(t 2+ ) = T2(ti+ ), provided (fi is a solution of (6.12), and then (fi+ is also a solution (determined by the parameter a and the value
( nW) U = U+W,

(r2W)V = V+W,

6.3. Discrete 2D integrable systems on graphs

215

which are satisfied identically in A, with the same matrix W as in (6.9). These equations assure the solvability of the system consisting of (6.16) and similar equations for the matrix function defined by (6.10). This latter matrix \I/+ is nothing but the extended frame of the transformed surface. Bianchi’s permutability theorem is formulated exactly as in the contin­ uous case, and is expressed by the same formula (6.11).

6.3. Discrete 2D integrable systems on graphs Before we turn to the explanation of the crucial idea that the 3D consis­ tency property of 2D equations should be taken as the definition of their integrability, we provide a bit more details on the notion of integrability, corresponding to the traditional view of integrable systems, which is based on discrete zero curvature representations. This latter notion works in a more general context than systems on a regular square lattice Z2, namely it is naturally formulated for systems on graphs. A graph 9 will mean for us not just a combinatorial object, but will be provided with an additional structure of a strongly regular polytopal cell decomposition of an oriented surface. The set of its vertices will be denoted by V($), the set of its directed edges, by E ( S), and the set of its faces, by F ( 9). To any such 9 there canonically corresponds a dual cell decomposition 9*; it is only defined up to isotopy, but can be fixed uniquely with the help of the Voronoi-Delaunay construction. The vertices of 9* are in a one-to-one correspondence with the faces of 9 (actually, they can be chosen as some points inside the corresponding faces; cf. Figure 6.2).

Figure 6 .2 . A face of S and the corresponding vertex of S*.

The variables of a discrete system (fields in the terminology of mathe­ matical physics) will be understood as elements / of some set X (the phase

216

6. Consistency as Integrability

space of a system), assigned either to the vertices or to the edges of S- (One can imagine also a mixed situation, where part of fields are assigned to the vertices and the others to the edges.) The system itself will be of the follow­ ing nature. Consider a closed path of directed edges which constitute the boundary of a face of Sti = ( x i , x 2),

C2 = (ar2 ,X 3 ),

tn = {xn, xi ).

Then, in the case of fields assigned to the vertices, it is supposed that the fields f ( x i ) , . . . , f ( x n) satisfy a certain condition, of a geometric or an ana­ lytic nature, called the equation associated to the face: (6.20)

Q ( / ( x i ) , . . . , / ( x n) ) = 0 .

If the fields are assigned to the edges, / ( e i ) , . . . , / ( e n), then the equation should read correspondingly: (6.21)

Q(/(ei),...,/(e„))=0.

A discrete system is a collection of such equations associated with all faces of S. One says that such a system admits a discrete zero curvature representa­ tion if there is a collection of matrices L(e; A) G G[A] from some loop group G[A], associated with every directed edge e G E ( S) (so called transition ma­ trices), with the following properties. For a system with fields on vertices, L(e; A) depends on the fields / ( x i ) , / ( a ^ ) if e = (£ 1, ^ 2); f°r a system with fields on edges, L(e; A) depends just on the field /(e). The argument A of the loops from G[A] is known in the theory of integrable systems as the spectral parameter. It is required that:

• for any directed edge e = (xi,#2)> if — (6.22)

then

L( -e,A) = (L(e,A))“ 1; • for any closed path of directed edges ei = {xi,x2},

e2 = ( x 2 , x 3 ),

...,

en = (xn, xi ),

we have (6.23)

L(tn, A) •••L(e2, A)L(ei, A) = 1.

In the case when the path bounds a face of 9, the discrete zero curvature condition (6.23) must be equivalent to (or at least a consequence of) the equation for the corresponding face. Under conditions (6.22), (6.23) one can define a wave function ^ : V(9) —> G[ A] on the vertices of S, by the following requirement: for any

6.4. Discrete Laplace type equations

217

directed edge e = ( x i , ^ ) £ ^(S ), the values of the wave functions at its ends must be connected via (6.24)

®( x 2,A) = L( c,A) ^( x i ,A).

For an arbitrary graph, the analytical consequences of the zero curvature representation for a given collection of equations are not clear. However, in the case of regular graphs, such as those generated by the square lattice Z + iZ C C, or by the regular triangular lattice Z + e27™/3Z C C, such a representation may be used to determine conserved quantities for suitably defined Cauchy problems, as well as to apply powerful analytical methods for finding concrete solutions.

6.4. Discrete Laplace type equations There exist discrete equations on graphs which are not covered by the con­ structions of Section 6.3.

Definition 6.1. (Discrete Laplace type equations) Let 9 be a graph, with the set of vertices V^(9) and, the set of edges E(S).

Discrete Laplace

type equations on the graph 9 for a function f : F ( 9 ) —* C read:

(6.25)

^

(f(xo),f(x);v(xo,x)) = 0.

xE star(xo)

There is one equation for every vertex x q G V( 5 ) ; the summation is extended over star(xo )7 the set of vertices of 9 connected to

xq

by an edge (see Figure

6.3/; the function
v (x 0 , x ) ( f ( x ) - f ( x 0)) = 0 , xE star(xo)

with some weights v : E ( 5 )

R+ assigned to the (undirected) edges of 9-

The notion of integrability of discrete Laplace type equations is not well established yet. We discuss here a definition which is based on the notion of the discrete zero curvature representation and works under an additional assumption about the graph 9 - Namely, like in the previous section, it hats to come from a strongly regular polytopal cell decomposition of an oriented surface. W e consider, in som ewhat more detail, the dual graph (cell decom po­ sition)

9 *-

E ach e G E ( 5 ) separates two faces of

spond to two vertices of

9 *-

9,

which in turn corre­

A path between these two vertices is then

declared the edge e* G E($*) dual to e. If one assigns a direction to an edge

6. Consistency as Integrability

218

X2

Figure 6.3. Star of a vertex xq in the graph

Figure 6.4. Face of

S-

of S-

S* dual to a vertex xo

e G E ( 9), then it will be assumed that the dual edge e* G E ( 9*) is also directed, in a way consistent with the orientation of the underlying surface, namely so that the pair (e,e*) is positively oriented at its crossing point. This orientation convention implies that e** = —e. Finally, the faces of 9* are in a one-to-one correspondence with the vertices of 9: if xo G V"(S), and x i , . . . , x n G V'(S) are its neighbors connected with xo by the edges * i — • • • ? — ( x o ,X n ) G E ( 9), then the face of 9* dual to x q is bounded by the dual edges ej = (yi, 2/2)? •••>cn — (Un, 2/ 1); see Figure 6.4. We will say that a system of discrete Laplace type equations on 9 pos­ sesses a discrete zero curvature representation if there is a collection of ma­ trices L(e*;A) G G[\] from some loop group G[A], associated to directed edges e*

G

E ( 9*) of the dual graph 9*, such that:

• the matrix L(e*; A) depends on the fields f ( x 0), f ( x) at the vertices of the edge e = (xq,x) G £?(9), dual to the edge e* G £(9*), and • the flatness conditions (6.22), (6.23) on the dual graph are satisfied. T h e m atrix L(e*;A) is interpreted as a transition m atrix along the edge

e*

G E ( S * ), th a t is, a transition across the edge

e

G E (S )-

T h e wave

function ^ in this situation is defined on the set V (S *) of vertices of the dual graph.

6 .5 . Q u ad -grap h s Although one can consider 2D integrable systems on very different kinds of graphs on surfaces, there is one kind — quad-graphs — supporting the most fundamental integrable systems.

6.5. Quad-graphs

219

Definition 6.2. (Quad-graph) A quad-graph D is a strongly regular polytopal cell decomposition of a surface with all quadrilateral faces. Since we are interested m ainly in the local theory of integrable system s of quad-graphs, and in order to avoid the discussion of some subtle boundary effects, we shall always suppose th a t the surface carrying the quad-graph has no boundary. Q uad-graphs are privileged because from an arbitrary strongly regular p o lytop al cell decom position called the double of

9

one can produce a certain quad-graph D ,

9-

T h e double D is a quad-graph, constructed from

9

and its dual

9*

as

follows. T h e set of vertices of the double D is V(*D) = F ( 9 ) U 1^(9 *)- Each pair of dual edges, say e =

(x 0, x i )

defines a quadrilateral (xq , j/i, x\, y2)-

G

25( 9 )

and e* =

(2/1, j/2) € E ( 9 *),

These quadrilaterals constitute the

faces of a cell decom position (quad-graph) D .

Thus, a star of a vertex

£ V ( 5 ) generates a flower of adjacent quadrilaterals from F ( D ) around #o; see Figure

6 .5.

L et us stress th a t edges of V belong neither to

25(9 )

nor

to E ( 9*).

X2

Figure 6 .5 . Faces of D around the vertex

xq .

Quad-graphs D coming as doubles are bipartite: the set F(D) may be decomposed into two complementary halves, V^D) = Vr(9)UVr(9*) ( “black” and “white” vertices), such that the ends of each edge from E(T>) are of different colors. Equivalently, any closed loop consisting of edges of D has an even length. The construction of the double can be reversed. Start with a bipartite quad-graph D. For instance, any quad-graph embedded in a plane or in an open disc is automatically bipartite. Any bipartite quad-graph produces two dual polytopal (in general, no more quadrilateral) cell decompositions 9 and

6. Consistency as Integrability

220

9*, with F(9) containing all the “black” vertices of D and V (9*) containing all the “white” ones, and edges of 9 (resp. of 9*) connecting “black” (resp. “white” ) vertices along the diagonals of each face of D. The decomposition of V(T>) into V ( 5 ) and F(9*) is unique, up to interchanging the roles of 9 and 9*. Notice that if a quad-graph D is not bipartite (i.e., if it admits loops consisting of an odd number of edges), then one can easily produce from D a new even quad-graph D', simply by refining each of the quadrilaterals from F(*D) into four smaller ones. Since we are interested mainly in the local theory, we always assume (without mentioning it explicitly) that our quad-graphs are cellular decom­ positions of an open topological disc. In particular, our quad-graphs D are always bipartite, so that 9 and 9* are well defined.

6.6. Three-dimensional consistency An attentive examination of examples in Sections 6.1, 6.2 leads to remark­ able observations which relate to the main philosophy of this book. For the continuous sine-Gordon equation the theory seems to consist of several com­ ponents of a rather different nature: the main object is a partial differential equation, its Backlund transformations are described by a compatible sys­ tem of two ordinary differential equations, while the superposition formula of Backlund transformations is expressed in purely algebraic terms. In the discrete context situation changes dramatically. All components of the discrete theory have essentially one and the same structure: equation (6.12) which describes discrete K-surfaces, equations (6.17), (6.18) for Backlund transformations of discrete K-surfaces, and equa­ tion (6.11) for the superposition principle of the latter. Their common struc­ ture is captured in the following formula for a function 0 : Zm —> M on an m-dimensional lattice: (6.27)

sin j

4

+ k + fa - fa). aj

4

Here the subscript j stands for the shift in the j-th lattice direction, and parameters a j are assigned to all edges parallel to the j-th lattice direction. Actually, in the geometric context, we are dealing with the case m — 4. The subscripts 1,2 label the coordinate directions of the discrete surfaces, while the subscripts 3,4 are used as replacements of the Backlund superscripts (1),(2). The relevant values of the parameters are: a\ — e/2, a 2 = 2/e, c*3 = a, and a± = (3. Equations (6.17), (6.11) are exactly of the form (6.27), and equations (6.12), (6.18) are brought into this form upon changing

221

6.6. Three-dimensional consistency

the sign of (u) —> (—1)U2(f)(u). This reflects the fact that the underlying geometric properties of dis­ crete K-surfaces and their Backlund transformations are identical and are captured in the definition of multidimensional K-nets, i.e., A-nets (nets in M3 with planar vertex stars) satisfying the additional requirement that in every elementary quadrilateral the opposite sides have equal length. A dis­ crete K-surface is a K-net with m = 2, iterated Backlund transformations of a discrete K-surface form a K-net with m — 3, while Bianchi’s permutability theorem for two Backlund transformations of a discrete K-surface deals with K-nets with m — 4. The variable transformation / = exp(z0/2) puts (6.27) into the form fa oo\

fjk _

(6 8)

T

a j f j ~ a kfk

“

which is also known as the Hirota equation.

fjk

Figure 6.6. 2D equation.

Oti

fijk

Figure 6.7. 3D consistency.

The Hirota equation (6.28) is a two-dimensional discrete equation, since it relates the variables / at the vertices of any elementary two-dimensional cell (square) of the m-dimensional lattice in such a way that any three variables determine the fourth one uniquely. The possibility to impose this equation everywhere on the m-dimensional lattice hinges on the case m = 3. The corresponding property of three-dimensional consistency should be un­ derstood as follows: suppose that four values / , f i , f j , f k are given (consult Figure 6.7 for notation). Then equation (6.28) defines fij , f j k and / ^ , and a further application of this equation gives three a priori different values of f^ k . These three values turn out to automatically coincide for arbitrary

6. Consistency as Integrability

222

initial data. Indeed, a direct computation shows: ,

(

lb. yj

, Jij k ~

<*k{<Xj - (xl)fifj + otiiotl - Oi))fjfk + a jia f - a 2 k) f kfi a k { a 2 _ a 2)fk + < a 2 _ a 2) f . + a j { a 2 _ a 2 )f]

■

This coincidence is the meaning of the 3D consistency of the Hirota equation. As a consequence, the Hirota equation can be consistently imposed on all elementary squares of a multidimensional lattice.

6.7. From 3D consistency to zero curvature representations and Backlund transformations Now we are in a position to expose the main idea concerning the under­ standing of discrete integrable systems, namely that the property of 3D consistency observed in Section 6.6 for the Hirota equation is actually of a fundamental importance and leads directly to the core of the whole theory. We show that other features of integrable systems, such as zero curvature representations and Backlund transformations, are consequences of 3D con­ sistency. The present section is devoted to a realization of this idea for systems on quad-graphs with fields on vertices and with labelled edges. A typical representative of this class of equations is the Hirota system, which we write here once more in the form (a on\

f 12

f

'

_ a i f l ~ a<2f 2 C*l/2 - <*2/l

’

In the geometric context of K-surfaces we had / = exp(i/2) G S1. In the present analytic study we will assume the fields / to be any complex numbers assigned to the vertices of Z 2, while ai are (complex) parameters naturally assigned to the edges of Z2 parallel to !B?; and constant along the strips in the complementary direction. In a different fashion, one can view ai as fields satisfying the labelling property (6.31)

S2a 1 = 0,

Sia2 = 0.

The Hirota system is 3D consistent, with (6.32)

/ 123 =

013(0% - « i ) / i /2 + ai(a§ - Oil)f2h + <*2(a? ~ al) f o h a 3(a| - a f ) / 3 + a i ( a § - aj )f i + a 2( a f - a | ) /2

One more example of such a system with vertex variables and edge parameters having the labelling property is given by the cross-ratio equation: (6.33)

q ( f , f i , f i 2 , f 2) = — a2

We already studied this system in Section 4.3 in the context of discrete isothermic surfaces, where the fields / are points of R N and parameters ai

6.7. From 3D consistency to zero curvature representations

223

are real numbers, the cross-ratio being defined according to the Clifford mul­ tiplication in G£(Rn ). Here we will consider a simpler version with fields and parameters being complex numbers. The commutativity of complex multi­ plication makes the check of the 3D consistency of the cross-ratio equation the matter of a straightforward computation, leading to ,f

,

=

123

(«1 - « 2 ) / l / 2 + (<32 - Qj3) / 2 /3 + (<*3 - Q = l)/3 /l (<*2 — « l ) / 3 + (« 3 _

0:2)f l

+ (a i — 0 's )f 2

A general system of this class consists of equations (6.35)

Q ( f , f i , f i 2 , f 2 ' , a i , a 2) = 0.

Here / : Z 2 —> C are complex fields, and a* are complex parameters on the edges of Z 2 parallel to 3 satisfying the labelling condition (6.31); see Figure 6.6. Actually, just from the outset we would like to generalize this setup by considering systems on arbitrary quad-graphs instead of Z2. In this case (6.35) should be read as a relation for fields / : V (D) —►C, with a : E(T>) —> C being a labelling of edges of D, i.e., a function taking equal values on any pair of opposite edges of any quadrilateral from F(D). In the context of equations on general quad-graphs, there are no distinguished coordinate directions; nevertheless it will be convenient to continue to use notation of (6.35), with the understanding that indices are used locally (within one quadrilateral) and do not stand for shifts into globally defined coordinate directions. So, / , / 1, / 12, can be any cyclic enumeration of the vertices of an elementary quadrilateral. Sometimes we will stress the absence of global coordinate directions by writing (6.35) in a different system of notation, using just a cyclic enumeration of vertices: (6.36)

Q ( / i , / 2, /3, /4; <*,/?) = 0;

see Figure 6.8.

h

h

h

h Figure 6.8. A face of a labelled quad-graph; fields on vertices.

224

6. Consistency as Integrability

For the very possibility to pose equation (6.36) on general quad-graphs, this equation should be uniquely solvable for any one of its arguments f i E C; therefore the following assumption is natural by considerations in that generality: Linearity. The function Q is a polynomial of degree 1 in each argu­ ment f i (multiaffine), with coefficients depending on the parameters

a, 0:

(6.37)

Q(/i,/2,/3,/4;a,/?) = ai(a,/?)/i/2/3/4 "I------ b a i 6 (a ,0 ).

For the Hirota equation (6.30) one can take Q = a/ 1/2 + 0 /2/3 - a/ 3/4 ~ 0/i/4,

while for the cross-ratio equation (6.33) with complex-valued arguments one can take Q = 0(/i - /2X /3 - U ) ~ ot(f2 - /3X /4 - f i ) -

Assume now that equation (6.35) possesses the property of 3D consis­ tency; cf. Figure 6.7. We will demonstrate that this remarkable property automatically leads to two basic structures associated with integrability in the soliton theory: Backlund transformations and zero curvature represen­ tation.

Theorem 6.3. (3D consistency yields Backlund transformations) Let equation (6.35) be 3D consistent. Then f o r any solution f : V (D ) —►C of the corresponding system (6.36) on a quad-graph T), there is a two-parameter family of solutions /+ : V(*D) —> C of the same system, satisfying

(6.38) fo r all edges (/, f i ) E E ( T )) . Such a solution f + is called a Backlund trans­

form of f , and is determined by its value at one vertex of T> and by the parameter A.

Proof. We formally extend the planar quad-graph D into the third di­ mension. For this aim, consider the second copy of D and add edges connecting each vertex / E V(T>) with its copy f + E V ( V + ). (We slightly abuse the notation here, by using the same letter / for vertices of the quadgraph and for the fields assigned to these vertices.) On this way we obtain a “3D quad-graph” D, with the set of vertices

V (D ) = F (H )U F (K + ), with the set of edges E ( D ) = E ( V ) U £ ( D + ) U { ( / , / + ) : / € V ^ D )} ,

6.7. From 3D consistency to zero curvature representations

225

and with the set of faces F ( D ) = F ( V ) U F (D + ) U {(/ , f u f t , f + ) : /, /i € V ( V ) } .

Elementary building blocks of D are combinatorial cubes (/ ,/ i,/ 12,/ 2 > /+ >f i j /12’ f ‘t ) i as shown in Figure 6.9. The labelling on J5(D) is defined in the natural way: each edge (/ + ,/ ^ ) G E ( D + ) carries the same label a* as its counterpart (/, f i ) G E ^ D ), while all “vertical” edges (/, /+) carry one and the same label A. Clearly, the content of Figure 6.9 is the same as of Figure 6.7, up to notation. Now, a solution f + : V ( V + ) —> C on the first floor of D is well defined due to the 3D consistency, and is determined by its value at one vertex of and by A. We can assume that /+ is defined on V(*D) rather than on since these two sets are in a one-to-one correspondence. □

Figure 6.9. Elementary cube of D.

Theorem 6.4. (3D consistency yields zero curvature representa­ tion) Let equation (6.35) be 3D consistent. Then the corresponding system (6.36) on a quad-graph V admits a zero curvature representation with spec­ tral parameter dependent 2 x 2 matrices: there exist matrices associated to directed edges o f V ,

(6.39)

L(e, a(e); A) : E ( D ) -► G L(2,C)[A],

such that fo r any quadrilateral face (/, / 1 , / 12 , f 2)

(6.40)

G

F ( D) the equality

L ( f i 2 , f i , a 2; A)L(/i, /, a i; A) = L (/ i2, f 2, o t i ; \ ) L ( f 2y /, a 2; A),

holds identically in X.

Proof. Due to the linearity assumption, equations (6.38) can be solved for /+ in terms of a Mobius (linear-fractional) transformation of /+ with coefficients depending on /, ff. (6-41)

/+ = L( f i , f , oc i ; X) [ f + \.

226

6. Consistency as Integrability

Here we use the standard matrix notation for the action of Mobius trans­ formations: (6.42)

L[z] = ( az + b)(cz + d )~ l ,

where

L = ^ ^

'

Now 3D consistency for f^2 yields that for any /+ , (6.43) ^ £ (/ i 2, /i, a 2; A) L ( f u /, c*i; A) [/+] = L (/ 12, /2, a i; A )L(/2, /, a 2; A) [/+]. Therefore, (6.40) holds at least projectively, i.e., up to a scalar factor. A normalization of determinants of L (or any other suitable normalization) allows one to achieve that (6.40) holds in the usual sense. □ As an example, we derive a zero curvature representation for the Hirota equation (6.30). It will be convenient to redefine the spectral parameter in this case by A «—> A-1 . Then the Hirota equation on the vertical faces of Figure 6.9 is written as follows: ,+

,

f t = f-

f + - Xocifi f i - Aa j +

This is written as a Mobius transformation (6.41) with the matrix / Aoc{

- A OLiffi fi

Normalizing the matrix so that its determinant be constant (not depending on field variables), we arrive at (6.44)

a j ( / / i ) - ,/2

{fi/f)m

)■

A gauge transformation L ( f i , f, <*; A) -> A ~ 1( f i ) L ( f i , /, tti; A) A ( f ) ,

A (f) =

f l/2

o

0 r

m ) '

results in an alternative form of the transition matrices: (6.45) The above gauge transformation corresponds to the interpretation of matri­ ces (6.45) as describing the Mobius transformations of shifted variables: fi~/fi = L ( f i , f , a,; A) [/+//].

Transition matrices (6.45) essentially coincides with those from (4.30) (to identify both formulas, one has to replace in (6.45) the spectral parame­ ter A by iA, to redefine the parameters a* i—►tan(a?-/2), and to make an exponential change of field variables / = e” ^ 2).

227

6.8. Boundary value problems for integrable 2D equations

Our second example will be the derivation of the zero curvature repre­ sentation for the complex cross-ratio equation (6.33). Again, we redefine the spectral parameter by A i—> A-1 , so that equations on the vertical faces of Figure 6.9 read: u t - n u - n )

(/+- /)(/*- ft)

x

This gives the Mobius transformation (6.41) with (6.46)

L( f i , /, ai, X) = I + - 0 j - ( *

.

The determinant of this matrix is constant (equal to 1 — Ac^); therefore no further normalization is required. A more usual form of the transition matrices of the zero curvature representation for the complex cross-ratio equation is obtained by the gauge transformation L ( f i , f , a i ; \ ) ~ A - 1( f i ) L { f i , f , a i ; \ ) A { f ) ,

A(f) = ( j

,

which leads to the matrices (6.47)

f ~l

These matrices (6.47) are interpreted as matrices of the Mobius transforma­ tions acting on the shifted quantities: f t ~ f i = L ( f i , f , a t ; A) [ / + - / ] .

To summarize: 3D consistency of 2D quad-equations with complex fields on vertices and with labelled edges implies the existence of Backlund trans­ formations and of the zero curvature representation. This is not a pure exis­ tence statement but rather a construction: both attributes can be derived in a systematic way starting with no more information than the equation itself, they are in a sense encoded in the equation provided it is 3D consistent.

6.8. Geometry of boundary value problems for integrable 2D equations There are several important aspects of the problem of embedding of a quadgraph into a regular multidimensional square lattice, related to integrable equations.

228

6. Consistency as Integrability

6 .8 .1 . Initial value problem. We start with the question of correct initial value (Cauchy) problems for discrete 2D equations on quad-graphs. Let P be a path in the quad-graph 2), i.e., a sequence of edges tj = ( x j , x j + 1) G E ( V ) . We denote by E ( P ) = { t j } and V ( P ) = { x j } the set of edges and the set of vertices of the path P , respectively. One says that the Cauchy problem for a path P is well posed if for any set of data f p : V ( P ) —> C there exists a unique solution / : V ( V ) —> C such that f \ v ( P ) = fp - It is n°t difficult to find examples of paths on the square lattice for which the Cauchy problem is well posed. The task we are interested in is to characterize, for a given quad-graph D, all paths with this property.

Figure 6.10. One-corner

Figure

initial path.

initial path.

6.11. Staircase

A solution of this problem can be given with the help of the notion of a strip in D.

Definition 6.5. (Strip) A strip in V is a sequence of quadrilateral faces qj G F(T>) such that any pair qj~\, qj is adjacent along the edge tj =

i

n qj, and tj, t j + i are opposite edges of qj.

is a path in D* consisting of edges e* = ( q j - \ , q j )

G

In other words, a strip E(T>*) such that any

consecutive pair t*, e*+1 enter and leave the quadrilateral qj along a pair of opposite edges tj, t j + \. The edges tj are called traverse edges of the strip.

So, in a labelled quad-graph © any strip may be associated to a label a sitting on all its traverse edges tj. The strips come to replace coordinate

directions in a regular square lattice, and can be considered as a discrete ana­ log of characteristics for hyperbolic systems of partial differential equations with two independent variables.

Theorem 6 .6 . (W ell posed Cauchy problems on quad-graphs) Let V be a finite simply connected quad-graph without self-crossing strips, and let P be a path without self-crossings in V . Consider a 3D consistent equation of the type (6.35) on the quad-graph V . Then:

6.8. Boundary value problems for integrable 2D equations

229

i) I f each strip in V intersects P exactly once, then the Cauchy prob­ lem f o r P is well posed.

ii) I f some strip in V intersects P more than once, then the Cauchy problem f o r P is overdetermined (has in general no solutions).

iii) I f some strip in D does not intersect P , then the Cauchy problem f o r P is underdetermined (has in general more than one solution).

Proof. We shall only sketch the proof of the claim i). It is based on an embedding T of V ( V ) into the unit cube of Z n, where n is the number of edges in P (the number of distinct strips in D). Choose any vertex xo E V'(D ), and set T ( x o) = 0 E Z n. The image of any other vertex x E V(T>) is defined recurrently along a path connecting xq to x with the help of the following rule: For any two neighbors x , y E V(T>), if the edge (x ,y ) E E(*D) belongs to the strip number i E {1 ,2 ,..., n }, then T ( y ) = T ( x ) + ei (mod 2 ), where e* is the i-th coordinate vector of Z n. The result does not depend on the path connecting x to xo, since any closed path has an even intersection index with any strip; therefore contribution of any strip to T along a closed path vanishes. Edges and faces of D correspond to edges and two-faces of the unit cube in Z 71. The T-image of the path P is the path ( 0 , 0 , 0 , , 0 ), ( 1 , 0 , 0 , . . . , 0 ), ( 1 , 1 , 0 , . . . , 0 ), ... , ( 1 , 1 , 1, . . . , 1 ). It is clear that for a 3D consistent equation the data along this path define a well-posed Cauchy problem for the unit cube in Z n. In particular, these data uniquely determine the values of the solution on T ( V ( V ) ) . □ It should be mentioned that this theorem is not valid for equations with­ out the 3D consistency property. The next theorem is based on the zero curvature representation with a spectral parameter; therefore it is also spe­ cific for 3D consistent equations on labelled quad-graphs. We will formulate this theorem for a concrete equation (cross-ratio equation), but actually it applies under much more general circumstances. See, however, Exercise 6.5, illustrating an instance where this theorem is not valid.

Theorem 6.7. (Relating data on two Cauchy paths) Consider a generic solution of the cross-ratio equation (6.33) on a simply connected Let each of the two paths P = (xo, x i , . . ., x n) and P — (xo, x i , . . ., x n) in V with a common starting point xo = xo and a common end point x n — x n intersect each strip in 2) exactly once. Then the the fields (/o, / i, . . ., f n) along P determine the fields (/o, / i,. . . , f n) along P uniquely, as soon as the sequences of labels = a (x i_ i,x i) along P and quad-graph T>.

dii — a ( x i - i , x i ) along P are known, that is, without knowing any additional information on the combinatorics of V .

6.

230

Consistency as Integrability

Proof. The proof is based on the zero curvature representation of the cross­ ratio equation with the transition matrices L given in (6.47). By the hy­ pothesis of the theorem, the sequence ( d i , . . ., d n) is a permutation of the sequence ( a i , . . ., a n). Prom the zero curvature condition (6.23) it follows that (6.48)

' [ [ L ( f i , f i - u a i -,\) = l [ L ( f i J i- i , a i -,X).

Generally speaking, such an equality does not hold automatically for non­ normalized transition matrices, but in our case det L(/i, /;-i, a*; A) = 1 — Xcti, which yields the equality of determinants of the both sides of (6.48). Denote the left-hand side of (6.48) by T(X) = H L V i J i - u a n X ) .

All entries of this matrix are polynomials in A. We want to show that this matrix can be uniquely refactorized as *r\

r (A ) = U L ( / i ,/i_ 1, o i ;A), with /o = /o, f n — /n, and with a prescribed permutation (d*) of the parameters (a*) along the path P . We show that there is a unique matrix of the form

VO such that all entries of the matrix T (A )L _ 1(/i,/o,c*i; A) are polynomials in A. Since d etT (A ) = — Aa»), the points A = a ^ 1 are exactly those where T ( A) is degenerate. For a generic solution, rank T ( a J l ) = 1, so that d im k e rT (a “ 1) = 1. Define f\ by kerT(o;J”1) = M

~ ^

(recall that /o = /o)- Then the elements of the vector

are polynomials divisible by 1 — Adi. Now observe that

which immediately implies that T (A )L _ 1(/i, /o, d i ; A) is a polynomial in A. An inductive application of this procedure yields the desired refactoriza­ tion of the matrix T ( A). It remains to show that for the so found sequence

6.8. Boundary value problems for integrable 2D equations

231

( f i ) , we have f n = f n . But this follows immediately from the fact that the free term of the (12) entry of T ( A) is equal to /o —f n = /o —f n - This finishes the proof. □

This theorem has rather surprising consequences. Consider a quad-graph obtained from the regular square lattice by replacing some m x n rectangle by a finite simply connected quad-graph with the same boundary vertices. The resulting quad-graph is called a regular square lattice with a localized defect. We say that a defect is weak if all strips entering the defect leave it in the same direction, possibly in a different order. Figure 6.12 illustrates a weak 3 x 2 defect.

C*2

0:3

Oi\

Figure 6.12. A weak localized defect in the regular square lattice.

Consider a Cauchy problem for the cross-ratio equation on a regular lat­ tice with a weak defect, with the initial data outside of the defect. Suppose that all horizontal edges outside of the defect carry the same label a and all vertical edges outside of the defect carry the same label 0 (so that in Figure 6.12 there should be ct\ = a 2 — as). Compare the solution of this problem with the solution of the same Cauchy problem but on the regular square lattice without defects. Surprisingly, as a consequence of Theorem 6.7, the solutions will coincide outside of the defect. One can say that for the cross-ratio equation (and the likes) with a homogeneous labelling the weak defects are transparent. 6 .8 .2 . Extension to a multidimensional lattice. The problem of em­ bedding of a quad-graph 2) into a regular multidimensional cubic lattice has also aspects of a different flavor.

232

6. Consistency as Integrability

Theorem 6 .8 . (Rhombic embedding) A quad-graph D admits an em­ bedding in C with all rhombic faces if and only if the following two conditions are satisfied:

i) No strip crosses itself or is periodic. ii) Two distinct strips cross each other at most once. For a proof of this theorem we refer the reader to Kenyon-Schlenker (2004). We will show that rhombic embeddings are closely related to 3D consis­ tency of equations on D. Given a rhombic embedding p : F (D ) —►C with edges of unit length (which can always be achieved by scaling and will be assumed from now on), one defines the following function on the directed edges of D with values in S1 — {9 G C : \0\ = 1}:

6 49)

( .

6{x, y) = p (y) - p {x ),

V(x, y) € E ( D ) .

This function can be called a labelling of directed edges, since it satisfies 9 (—c) — —0(e) for any e G E ( V ) , and the values of 6 on two opposite and equally directed edges of any quadrilateral from F (D ) are equal. See Figure 6.13. For any labelling 9 : E(T>) —> S 1 of directed edges, the function 92 : E ( D ) —» S 1 is a labelling of (undirected) edges of V in our usual a - (fl sense.

/ Figure 6.13. Labelling of directed edges.

Definition 6.9. (Quasicrystallic rhombic embedding) A rhombic em­ bedding p : V'(D) —> C of a quad-graph D is called quasicry stallic i f the set of values of the function 9 : £?(D) —> S 1 defined by (6.49) is finite, say & = { ± 9 i , . . . , ±9d}.

An example of a quasicrystallic (actually periodic) rhombic quad-graph with d = 3 is the so-called dual kagome lattice shown in Figure 6.14. A prototypic example of a nonperiodic quasicrystallic rhombic quadgraph with d = 5 is the famous Penrose tiling shown in Figure 6.15.

6.8. Boundary value problems for integrable 2D equations

233

Figure 6.14. D ual kagome lattice.

Figure 6.15. Penrose rhombic tiling.

It is of a central importance that any quasicrystallic rhombic embedding p can be seen as a sort of projection of a certain two-dimensional subcomplex (combinatorial surface) of a multidimensional regular square lattice Z d.

The vertices of are given by a map P : V(T>) —+ Z d constructed as follows. Fix some x q G V(T>), and set P ( x o ) = 0. The images in Z d of all other vertices of D are defined recurrently by the property: For any two neighbors x, y G V (D ), if p (y ) — p (x ) = ±0* G ©, then P ( y ) — P { x ) — ± e * , where e* is the i-th coordinate vector of Z d. The edges and faces of the combinatorics of

correspond to the edges and faces of D, so that is that of D.

234

6. Consistency as Integrability

To exploit possibilities provided by the 3D consistency, we extend the labelling 9 : E(*D) —►S1 to all edges of Z d, assuming that all the edges parallel to (and directed as) e& carry the label Oj^. This gives, of course, also the labelling a — 62 of undirected edges of Z d. Now, any 3D consistent equation can be imposed not only on but on the whole of Z d: (6.50)

Q(/, f j , f jk , fk\<Xj,OLk) = 0 ,

1 < j ^ k < d.

Here indices stand for the shifts into the coordinate directions. Obviously, for any solution / : Z d —> C of (6.50), its restriction to V (f^x>) ~ V (D ) gives a solution of the corresponding equation on the quad-graph 2). As for the reverse procedure, i.e., for the extension of an arbitrary solution of (6.36) from D to Z d, more thorough considerations are necessary. An elementary step of such an extension consists in finding / at the fourth vertex of an elementary square from the known values at three vertices according to (6.50). Due to 3D consistency this extension is well defined. In particular, one can find / at the eighth vertex of an elementary 3D cube from the known values at seven vertices; see Figure 6.16. This can be alternatively viewed as a flip (elementary transformation) on the set of rhombically embedded quadgraphs 2), or on the set of the corresponding surfaces in Z d. Any quadgraph 2) (or any corresponding surface Qx>) obtainable from the original one by such flips, carries a unique solution of (6.50) which is an extension of the original one. /23

/23

/2

/2

fl2

fl2

fl F i g u r e 6.16. Elementary flip.

D e fin itio n

6 . 10 .

(H u ll) For a given set V C Z d, its hull J i ( V ) is the

minimal set J~C C Z d containing V and satisfying the condition: if three vertices of an elementary square belong to *H, then so does the fourth vertex.

One shows by induction that for an arbitrary connected subcomplex of 7Ld with the set of vertices V , its hull is a brick (6.51)

n a,6 = { n = ( n i , . . . , n d) € Z d : ak < n k < b k, k = l , . . . , d } ,

235

6.9. 3D consistent equations with noncommutative Reids

where (6.52)

ak = ak (V) = m in n k, nev

bk = bk(V) = max nk, nev

k = l,...,d ,

and in the case that n k are unbounded from below or from above on V, we set d k (V ) — —oo, resp. bk(V) = oo. Combinatorially, all points of the hull 9-C(F(f2x>)) can be reached from by the extension procedure described above. However, there might be obstructions for extending solutions of (6.36) from a combinatorial surface (two-dimensional subcomplex of Z d) to its hull, having nothing to do with 3D consistency. For instance, the surface fl shown in Figure 6.17 supports the solutions of (6.36) which cannot be extended to the solutions of (6.50) on the whole of J { ( V (fi)): the recursive extension will lead to contradictions. The reason for this is nonmonotonicity of fi: it contains pairs of points which cannot be connected by a monotone path in f!, i.e., by a path in fi with all directed edges lying in one octant of Z d. However, such surfaces f2 do not come from rhombic embeddings, and in the case of f ^ there will be no contradictions.

6 11

. . (E x te n s io n o f so lu tio n s fro m q u a d -su rfa c e s t o Z d) Let the combinatorial surface f d in Z d come from a quasicrystallic rhombic embedding of a quad-graph D, and let its hull be ^ ( V ^ d ) ) = n a,b- An arbitrary solution of a D consistent equation (6.36) on JId can be uniquely extended to a solution of equation (6.50) on n a?fc.

T h eorem

2

3

The proof of this theorem can be found in Bobenko-Mercat-Suris (2005).

6.9. 3D consistent equations with noncommutative fields The validity of the message formulated in the last paragraph of Section 6.7, saying that 3D consistency of a quad-equation yields a construction of Backlund transformations and of the zero curvature representation, is by no

236

6. Consistency as Integrability

means restricted to the situation for which it was demonstrated (complex fields on vertices). In the present section, we show that it can be extended to equations with fields on vertices taking values in some associative but noncommutative algebra A with unit over a field 3C, and with edge labels with values in X . The transition matrices of the zero curvature representation are in this case 2 x 2 matrices with entries from A . They act on A according to (6.42), where now the order of the factors is essential. Actually, the proof of Theorem 4.26 given in Section 4.3.7 is an example of a derivation of a zero curvature representation for the cross-ratio equation (6.33) with fields in A = C£(Rn ) and with parameters a* from X = E, which governs discrete isothermic surfaces in E ^ (one has to interpret the arbitrary parameter 0:3 in equation (4.96) as the spectral parameter A). The literal generalization of this proof for an arbitrary associative algebra A leads to the following statement.

Theorem 6.12. (Cross-ratio equation in an associative algebra) The cross-ratio equation in an associative algebra A is 3D consistent. I t possesses a zero curvature representation with transition matrices (6.47), where the inversion is treated in A .

We provide here two more examples of similar results for equations with noncommutative fields. The first will be dealing with the noncommutative Hirota equation. It turns out that the correct way to write such a noncom­ mutative generalization is the following: (6.53)

/12/-1 = (<*1 - a 2f 2 f i 1){oi\f 2 f i 1 - a 2) - \

Theorem 6.13. (Hirota equation in an associative algebra) The non­ commutative Hirota equation (6.53) is 3D consistent. It admits a zero cur­ vature representation with the transition matrices

(6-54)

A ) = ^ _ a .y_i

Proof. The noncommutative Hirota equation on the face (/, f i , f i j , f j ) can be written as a formula which gives fij as a linear-fractional transformation of /,■: (6.55)

f i j = ( aufi - a j f j ) ( a i f j - ay/i)-1 / = L(/<, /, on, a tj)[fj],

where (6.56)

L { f i , f , a h a j) = ^

a jf-^ fi

Here we use the same notation as in the proof of Theorem 4.26 given in Section 4.3.7 for the action of of the group GL(2,.A) on A . Thus, equation

6.9. 3D consistent equations with noncommutative fields

237

(6.53) on the faces Ci3, C23 of the elementary 3D cube C123 can be written as (6.57)

/13

=

L (/ i, / , a i, a 3)[/3],

(6.58)

/23

=

L(f2 ,f,a 2 ,<X 3 )[h }-

By the shift in the second, resp. the first, coordinate direction we derive the expressions for /123 obtained from equation (6.53) on the faces T2C13, T1C23, respectively: (6.59)

/123

=

£(/i2,/2,ai,a3)[/23],

(6.60)

/123

=

£(/i2,/i,a2,a3)[/i3].

Substituting (6.57), (6.58) on the right-hand sides of equations (6.60), (6.59), respectively, we represent the equality between the two values of /123 (which we want to demonstrate) in the following matrix form: £(/i2, /1, OL2, a 3)L (/ i, /, a i, a 3)[/3] = L (/ i2, /2,

a 3)L (/ 2, /, a 2, Of3)[/3].

In fact, a stronger claim holds, namely (6.61)

L(/i2, /1, a 2, a 3)£ (/ i, /, aa, a 3) = £(/i2, /2, a i, a 3)L (/ 2, /, a 2, a 3).

Indeed, the (11) entries on both sides of this matrix identity are equal to + a\a2f i 2f ~ l . Equating (12) entries on both sides is equivalent to the Hirota equation of the face (/, /1, /12, f 2), and the same holds for the (21) entries. Finally, equating the (22) entries is equivalent to the condition that /i2/ _1 commutes with f 2f i l , and this is, of course, true by virtue of (6.53). This proves the 3D consistency of the noncommutative Hirota equation. The claim about the zero curvature representation is nothing but relation (6.61) just proven with 0:3 replaced by A. □ We consider here one more equation of this kind: (6.62)

(/12 -

/ X / 2 - f i ) = ot2 - a i ,

with the vertex variables / taking values in A and with the edge labels a from X . In the case of real-valued fields /, this equation is known under the name of the discrete K d V equation; among other things, it expresses the Bianchi-type superposition formula for the Backlund transformations of the Korteweg-de Vries equation. In the case when the fields / are considered to belong to R N C A = Q£(RN ), the solutions of this equation are special T-nets in R N : (6.63)

/ 12 — / ~ | ; ; ; ^ 2 (/ 2 - / i ) .

In the vector form (6.63) this equation is known as the discrete Calapso equation.

238

6. Consistency as Integrability

Theorem 6.14. (Discrete K d V equation in an associative algebra) Equation (6.62) in an associative algebra A with unit is 3D consistent. It possesses a zero curvature representation with the transition matrices

(6.64)

L ( f i , f , a t ; A) = ( {

A"

ffl ) •

Proof. Equations (6.62) on the vertical faces of Figure 6.9 read: / + = / + (A - <*,)(/+ -

fi)-1= L(fi, / ,

a <; A) [/+].

This gives the transition matrices, which can then be used to prove the 3D consistency, in the same manner as in the proof of Theorem 4.26 given in Section 4.3.7 and in the proof of Theorem 6.13. □ Our last example in this section is of a geometric origin and of a slightly different nature than the previous ones. In Section 4.1, in our study of T-nets in quadrics, we encountered the equation with vertex variables / :

Z2 ^ Q = { / e l " : ( / , / ) = f (6.65)

f

nlf

4

f\ ~~

n

2 ( / , / l - / 2)

a = «? -U u k )=

Ih ~ k ?

■

A priori it does not contain any parameters. However, the quantities ai — 2 (/, f i ) , being functions of the vertex variables / rather than parameters of the equation, possess the labelling property (6.31). Comparing (6.65) with (6.63), we see that the former can be regarded as a particular instance of the latter.

Theorem 6.15. (T-nets in quadrics) Equation (6.65), describing T-nets in quadrics, is 3D consistent. I t possesses a zero curvature representation with transition matrices with entries from C£(RN ):

(6 -66 )

£ (/ i,/ ;A )= ( {

Xt j f ) -

Proof. 3D consistency has been proven geometrically in Theorem 4.3. As for the transition matrices, we can take those from (6.64) with

= 2 (fji) -

-ffc - fif .

Note the geometrical meaning of the spectral parameter: A = 2 ( f , f + ) for the Backlund transformation /+ from which the transition matrices are constructed. □

239

6.10. Classification of discrete integrable 2D systems. I

6.10. Classification of discrete integrable 2D systems with fields on vertices. I The notion of 3D consistency, being fundamental for definition and study of 2D integrability, proves extremely useful also in various classification prob­ lems of the integrable systems theory. Because of its constructive nature, it can be put into the basis of classification within certain Ansatze. We will present here the solution of a very general problem concerning the 3D con­ sistent systems of (possibly different) quad-equations with complex fields on vertices. Quad-equations will be of the form (6.67)

Q ( x i , x 2 ,x 3 ,x 4) = 0,

where the field variables X{ G C P 1 are assigned to the vertices of a quadrilat­ eral (ordered cyclically), and Q satisfies only one assumption, namely that of linearity , formulated already in Section 6.7: the function Q is supposed to be an irreducible polynomial of degree 1 in each variable. This implies that equation (6.67) can be solved for any variable, and the solution is a rational function of the other three variables. The problem we would like to solve is that of the 3D consistency of six a priori different quad-equations put on the faces of a coordinate cube: the system of six quad-equations,

(6.68)

A (X , X U X12,X2) = 0, B ( x , x 2, x 23,

z 3) = o,

A (X 3JX 13,^123^23) = 0, B ( x 1 12 123 13 0

C (x , £3, £ 23, Z i ) = 0 ,

C ( x 2, x 23, £123, X\2) = 0 ,

, £ ,^ ,^ ) = ,

should admit a unique solution £123 for arbitrary initial data £, £ 1, £ 2, £3 ; see Figure 6.18. The functions A , . . ., C are affine linear and a priori are not supposed to be related to each other in any way. In solving a classification problem, one should factor out a possibly large group of transformations that leave the class of objects being classified in­ variant. In our problem each quad-equation preserves its form under the group ( M o b ) 4 which acts by Mobius transformations on all the vertex fields independently. It will be convenient to denote by the set of polynomials in n variables which are of degree rn in each variable, with the following action of Mobius transformations on a polynomial P G CP™: M [ P ] ( x l , . . . , £ n)

= (c lX l + dx) m • • • (cnx n + dn) mp ( a i X i } ^ \Ci£i + cL\ where (6.69)

— aidi

biC{ 7^ 0 .

,

anXn + bn\ cnx n “j- cLji/

240

6. Consistency as Integrability

X23

£123

F i g u r e 6.18. A 3D consistent system of six different quad-equations: A and A are associated to the bottom and to the top faces of the cube, B and B , to the left and to the right faces, and C and C , to the front and to the back ones.

Thus, quad-equations (6.67) are characterized by polynomials Q G 7\. An important step in the solution of our problem will be classifying such poly­ nomials and finding their normal form modulo the action of (M ob)4. The full problem we are aiming at is classifying and finding normal forms for 3D consistent systems (6.68) modulo the action of ( M d b )s (independent Mobius transformations of all eight vertices of the cube). We will solve these problems under certain nondegeneracy conditions. In formulation of these conditions, as well as in the whole theory, the following operations play an important role: (6.70)

&Xi,Xj

: ^4

^2?

dxi,Xj(Q) — Q x i Q x j ~ QQ xiXj

•

(Variables placed as subscripts stand for partial differentiations.) The oper­ ation 5Xi,Xj applied to an affine linear polynomial Q ( x i, £ 2, #3> ^ 4) eliminates the variables Xi , Xj, the result being a biquadratic polynomial of the remain­ ing variables xk,xi (so that {z ,j, k , l } = {1 ,2 ,3 ,4 }), which we will denote by hkl(x k ,x i) = hlk(x k ,x i). Thus, from any Q G IP\ the operations SXiiXj produce six biquadratic polynomials hkl G 7 ^ f ° ur ° f them corresponding to the edges of the underlying quadrilateral, and the remaining two corre­ sponding to the diagonals. Note that the operations SXuXj are covariant with respect to Mobius transformations: (6.71)

5Xl,Xj( M [ Q } ) = A i A j M [ 6 x ilXj(Q )],

with A i given in (6.69).

6.10. Classification of discrete integrable 2D systems. I

241

Definition 6.16. (Nondegenerate biquadratic) A biquadratic polyno­ mial h ( x , y) E

is called nondegenerate i f no polynomial in its equivalence

class with respect to Mobius transformations is divisible by a factor x — c or y — c (with c = const).

Thus, a polynomial h ( x , y ) E 7 2 ls nondegenerate if it is either irre­ ducible or of the form (a\xy + (3\x + 71 y + 5 \)(ot 2xy + fa x + 722/+ <^2) with a{Si — 7^ 0. In both cases the equation h = 0 defines y as a two-valued function on x and vice versa. An example of a degenerate biquadratic is given by h (x, y) = x — y 2 (considered as an element of 7 2), since under the inversion x 1—►1 /x it turns into x ( l — xy 2).

Definition 6.17. (Quad-equation of type Q ) A multiaffine function or the corresponding quad-equation Q = 0, is said to be of type Q i f its fo u r accompanying edge biquadratics h^k E 7 2 are nondegenerate, and it is said to be o f type H otherwise.

It turns out that multiaffine equations of type Q admit an exhaustive classification modulo (M 06)4, with only four normal forms.

Theorem 6.18. (Classification of equations of type Q ) Any multi­ affine equation Q(x\, X2 , £ 3 , X 4 ) = 0 of type Q is equivalent, modulo Mobius transformations from ( M o b )4 acting on each of the variables independently, to one of the equations in the following list, called the list Q:

(Q4)

sn (ot)(xiX 2 + xzX 4 )-s ii((3 )(x iX 4 : + X2Xz)-sn(oL-(3)(xiXz-\-X2X4)

+ sn(a - (3) sn(a) sn(/?)(l + k2x iX 2^ 3^ 4) = 0, (Q3)

s m ( a ) ( x i x 2 + x s x 4 ) - s m ( p ) ( x i x 4 + x 2x s ) —s m ( a - p ) ( x i x s + x 2x 4:) + 6 sin(a — (3) sin(a) sin(j3) = 0 ,

(Q2)

a ( x i x 2 + X3X4 ) - P (x \ x 4 + X2 X3 ) - ( a - (3)(x 1 X3 + X2X4 ) +a/3 ( a - ( 3 ) ( x i + X 2 + X 3 + X 4 )-OLf 3 (a - / 3 ) ( a 2 - a l 3 +/3 2) = 0,

(Q l)

a ( x i x 2 + X3X4 ) - p ( x 1 X4 + X2 X3 ) - ( a - j3)(x 1X3 + # 2X4 ) +5af3(a — (3) = 0 .

In equation (Q 4 ) the notation sn(a) = sn(a; k) is used f o r the Jacobi elliptic function with modulus k. The parameter 5 in equations (Q 3), ( Q l ) can be scaled away, so that one can assume without loss of generality that S = 0 or 6

= 1.

It is important to observe that there were no a priori built-in parameters a, (3 in the polynomial Q E they appear in the course of classification. They turn out to be naturally assigned to the edges of the quadrilateral (x i,X 2 ,X 3 ,X 4). Equation (Q4) is the most general one of the list; it is parametrized by two points of an elliptic curve. Equations (Q 1)- (Q 3 ) are obtained from (Q4) upon degenerations of an elliptic curve into rational

242

6. Consistency as Integrability

curves. One could be tempted to reduce the list Q to one item (Q4). How­ ever, the limit procedures necessary for that are outside of our group of admissible (Mobius) transformations, and, on the other hand, in many sit­ uations the “degenerate” equations (Q 1)-(Q 3) are of interest by themselves (for instance, the simplest equation of the list, (Q l) with S = 0, is nothing but the complex cross-ratio equation). This resembles the situation with the list of the six Painleve equations and the coalescences between them. It remains to find out how the equations of Theorem 6.18 can be assem­ bled into 3D consistent systems on a cube.

Theorem 6.19. (3D consistent systems of type Q ) Each equation of the list Q is 3D consistent. Conversely, any 3D consistent system (6.68) with all six equations of type Q is equivalent, modulo Mobius transformations from (M d b )s acting on each variable independently, to the system

(6.72)

Q ( x , x u x i j , x j ; a i , a j ) = 0,

Q ( x k,Xik, xi2 3 ,xjk;0tu0tj) = 0,

where ( i j k ) stands f o r any of the three cyclic permutations of (123), and Q(x\, X2 , £3 , £4 ; a, (3) is one of the polynomials (Q l)-(Q J ^ ).

In the next section, we sketch the main ideas behind the proof of this remarkable result.

6.11. P ro o f of the classification theorem 6.11.1. 3D consistent systems, biquadratics and tetrahedron prop­ erty. Biquadratic polynomials hli for a given Q 6 7\ are closely related to the so-called singular solutions of the basic equation (6.67). Generically, the equation Q ( x 1,£ 2,^ 3 ,^ 4) — 0 can be solved with respect to any variable: if Q = p ( x j , x k , x i ) x i + q ( x j , x k, x i ) then X{ = —q/p for generic values of x j , x k,xi. However, Xi is not determined if the point ( x j , x k, x{ ) lies on the curve Si in (C P 1) 3 defined by (6.73)

Si :

p ( x j , x k, x t) = q( xj , x k, x t) = 0.

Since p = Q Xi and q = Q — Xip — Q —XiQ x%, equations (6.73) are equivalent to Q( XUX2, X3, X4) = Q Xi( x 1,X2,X3,Z4) = 0.

Definition 6.20. (Singular solution) A solution (x i, X2 , £3, £ 4 ) of equa­ tion (6.67) is called singular with respect to Xi i f it satisfies also the equation Q x i { x i , x 2 ,x z ,x ± ) — 0. The set of solutions singular with respect to Xi is parametrized by the curve (6.73) called the singular curve f o r Xi.

The projection of the curve Si onto the coordinate plane (fc, I) is exactly the biquadratic hkl = pqXj - pXjq = Q XtQxj ~ Q Q Xi,xj = 0-

243

6.11. Proof of the classification theorem

Lemma 6.21. (Singular solutions and biquadratics) I f a solution (X\,X 2 ,X 3 ,X 4 ) of equation (6.67) is singular with respect to Xi, then h^k = W 1 = hkl = 0 on this solution. Conversely, if hkl = 0 f o r some solution, then this solution is singular with respect to either X{ or X j .

Proof. Since hkl = Q XiQ Xj ~ QQxi,xj >the equation hkl = 0 on the solutions of the equation Q = 0 is equivalent to Q XiQ Xj = 0 -

O

In the following theorem we will deal with biquadratic polynomials cor­ responding to various multiafSne ones; we will denote the biquadratics by the same letters as their parent quad-equations, with the superscripts for the remaining variables, so that, e.g., A 0,1 = 5X2^Xl2A is the result of eliminating X2 ,X \2 from A ( x , x i , x i 2 , x 2)-

Theorem 6.22. (Tetrahedron property and biquadratics for 3D consistent systems) Consider a 3D consistent system ( 6 .68 ) with all six functions A , . . . , C being of type Q. Then:

• The system ( 6 .68 ) possesses the tetrahedron property: the value of x \23 as a function of the initial data x , x i , x 2 , x 3 does not depend on x ; see Figure 6.19.

• F or any edge of the cube, the two biquadratic polynomials corre­ sponding to this edge, coming from the two faces sharing this edge, coincide up to a constant factor.

Proof. The values of X123 obtained from the equations A = 0, B = 0 and C = 0 , respectively, result from elimination of £ 12, X13 and £ 23, which can be expressed by the equations 7-1 / 2

1

1

3

1

X

F(X,XI,X2,X3,X123) Axi3,x23^C ,

2

3

1

1

1

A X23B C Xi3

A X13B X23C + A B X23C Xl3

0,

B X13C A X12

B X12C XisA -f- B C X13A X12

0,

Cxi 2 -ABX23

C X23A Xi2B + C A X12B X23

0.

,

G {x ,x i,X2,ar3,xi23) B Xi 2,X\zCA u(2

1

3

1

1

\

H (X ,X UX2,X3,X123)

^12^23^

Here the numbers over the arguments of the polynomials F , G , H indicate their degrees in the corresponding variables. These degrees are in the projec­ tive sense, that is in agreement with the action of Mobius transformations, and can be read off the right-hand sides. Due to 3D consistency, the expres­ sions for £123 as functions of x , x i , x 2>^3 found from these three equations,

244

6. Consistency as Integrability

coincide. Therefore the polynomials F, G, H must factorize as:

F = f { x , x 3) K ,

G = g (x ,x i)K ,

H = h ( x , x 2) K ,

K = K ( x , x i,® 2 , £ 3 , 2123),

where the polynomial K yields the common value of x 123 as a function of x , x i , x 2,xs. Here f , g , h are some polynomials of degree 2 in the second argument. The degrees of f , g , h and K in x remain to be determined. We do this by analyzing singular solutions. Let the initial data x , x \ , x 2 be free variables, and let x% be chosen to satisfy the equation f ( x , x 3) = 0. Then F = 0, and thus the system B = C = A = 0 does not determine the value of £ 123. Moreover, the equation B = 0 can be solved with respect to x 2% since otherwise the initial data must be constrained by the equation B ° '2( x , x 2) = 0 . Analogously, the equation C = 0 is solvable with respect to £ 13. Therefore, the uncertainty appears from the singularity of equation A = 0 with respect to £ 123. Hence, the relation A 3,23(x 3 , 2:23) — 0 is valid. In view of the assumption of the theorem, #23 is a (two-valued) function of £3 and does not depend on x 2. This means that the equation B — 0 is singular with respect to x 2 and therefore B 0,3 (x ,x 3) = 0 . Analogously, (70,3 (x ,x 3) — 0 . Thus, we have proven that if x% = ip(x) is a zero of the polynomial / then it is also a zero of the polynomials B 0,3, C 0,3. If one of these three polynomials is irreducible, then this already implies that they coincide up to a constant factor. If the polynomials are reducible, then we could have / = a2, i ?0,3 = ab, C 0,3 = ac, where a, 6 , c are multiaffine in x, X3 . In any case, degx / = 2 , and this is sufficient to complete the proof. Indeed, this implies degx K = 0 , so the tetrahedron property is valid, and the first statement of the theorem is proven. In turn, the tetrahedron property can be used to prove the relation (6.74)

A o,1 5 0 ,2 ^ 0,3 + A 0,2B 0,3C 0,1 =

0

The variables in this relation separate: £0,3

yj0,l

£ 0,2

= ~ C ^ * AW ’ so that B 0,3/C 0^ may only depend on x. Due to nondegeneracy of the bi­ quadratics 5 ° ’3 ,C 0’3, this ratio is constant, which proves the second state­ ment of the theorem.

245

6.11. Proof of the classification theorem

So, it remains to prove equation (6.74). For this goal, rewrite the system ( 6 .68 ) in the form X12 = a(x, Xl, X 2 ),

£23 =

b (x, X 2 , X3),

£13 =

c(x,

X l , X3) ,

£123 = tt(£3,£l3>£23) = b(xi»£l2,£l3) — C(X2,X12,X23)Assuming the tetrahedron property, i.e., X123 = d (x i,x 2,x 3), we find by differentiation: '£1

^#13^X\ 5

dx 2

®JX23^X2 5

0 — ^Xi3^X + G,X23^X')

1 'X2

bx\2^X2 5

dx3

^X13^X3J

0 = bxi2ax + bx 13

'^3

CX23bxs5

dx 1 = cXl2aXl,

0 = cx23bx + CX12&X-

These equations readily imply the relation ^X2^X3^X\ "“t- ^JX\bx2^X'i

O’

It is equivalent to (6.74) in view of the identity aX2/aXl = A 0'1/A0’2.

□

F i g u r e 6.19. Tetrahedron property.

The astonishing tetrahedron property, possessed by all 3D consistent systems of type Q, is illustrated in Figure 6.19. It means that the fields #i ?# 2 ># 3 >£123 sitting at the vertices of the white tetrahedron are connected by a certain multiaffine relation K ( x 1, X2, X3, X123) = 0. Of course, for sym­ metry reasons, a multiaffine relation L ( x , X12, X23, X13) = 0 also holds for the fields at the vertices of the black tetrahedron. 6 . 1 1 .2 .

Analysis: descending from multiaffine Q to quartic r. In the

further analysis, one more operation similar to (6.70) will be useful, namely (6.75)

SXk: ? 2 ^ ? i ,

6Xk(h) = h2 X k-

2hhXkXk.

The operation ^Xk applied to a biquadratic polynomial h (x 1 ^x2 ) actually computes its discriminant with respect to the variable x k which gets elimi­ nated, the result being a quartic polynomial which we will denote by 77 (x/)

246

6. Consistency as Integrability

(where {£;,/} = {1 ,2 }). Thus, from any h G the operations SXk produce two quartic polynomials r/ G 7\. The operation SXk is covariant with respect to Mobius transformations: (6.76)

6Xl( M [ h ] ) = A 2M [5 Xl(h)}.

The following statement is proved by a straightforward computation.

Lemma 6.23. (Commutativity of discriminants) F or any multiaffine polynomial Q(x\, x 2, £ 3 , £ 4 )

G

5

SXk(SXi,Xj (Q)) — SXj ( Xi,Xk (Q)),

(6-77)

so that the following diagram is commutative: JX 3

r 4(x 4 )

UX4

ft34(£ 3 , £ 4 )

r 3( x 3)

JXi,X2

(6.78)

JX2,X3

h u (x ! , x 4)

UX4

JX2

Q ( x i , x 2,x 3 ,x 4)

J X \ ,£4

h23{x 2, x 3)

UX3

X 3 1X4

JX2

n (® i)

h 12(x i , x 2)

X\

T2{x 2)

In fact, this diagram can be completed by the polynomials h 13, hM cor­ responding to the diagonals (so that the graph of the tetrahedron appears), but we will not need them. Further on we will make an extensive use of relative invariants of polyno­ mials under Mobius transformations. For quartic polynomials r G IP4 these relative invariants are well known and can be defined as the coefficients of the Weierstrass normal form r = 4x3 — g2x — 53 . For a given polynomial r ( x ) = r±x4 + r$x3 + r 2x 2 + r\x + ro they are given by

9 2 (r ) = - ^ ( 2 r r IV g3( r ) =

2 r ' r ' " + ( r " ) 2) = ^ ( 12r 0r 4 - 3 rir 3 + r|),

^ ( 1 2 r r " r IV — 9( r ' ) 2r IV — 6r ( r ' " ) 2 + 6r ' r " r ' " — 2 ( r " ) 3) o40D

= ^ ( 7 2 r 0r2r4 - 27r2r4 + 9 rir 2r 3 - 27r 0r 3 - 2r f). Under the Mobius change of x = x\ these quantities are just multiplied by the constant factors: gk{ M [ r } ) = A f gk(r ) ,

k = 2,3.

247

6.11. Proof of the classification theorem

For a biquadratic polynomial h (6.79)

2,

h ( x , y ) = h 22X2y2+ h 2i x 2y + h 2ox2+ h i 2 xy2+ h n x y + h i o x + h o 2y2+ h o iy + h o o ,

the relative invariants are defined as ^2 (h ) ~ 2hhXxyy

2hxhxyy

2 hyhxxy + 2 hxxhyy “I- hXy

= 8 /l00^22 — 4/^01^21 — 4/^10^12 + 8 /l02^20 + ^ 11? ^

/ h

is{h) = - det I hy

hxy

hXx \ fh,22 hxxy j = det I h\2

h 21 hn

/l20 /iio

\hyy

hxyy

hXXyyJ

^01

^00

hx

\^02

Notice that i;j can be defined also by the formula - 4 i 3(h )

=

5x,y(8xty(h))/h.

Under the Mobius change of x = x\ and y — x 2 ,

ifc(M[/i]) = AfA^*fc(/i),

fc = 2,3.

The following properties of the operations 5XiV, 5X are proven straightfor­ wardly.

Lemma 6.24. (Opposite biquadratics and all four quartics have equal invariants) For any multiaffine polynomial Q{x\, x 2, £3, £ 4 ) E J 4 se£: /i12(x i,£ 2) = SX3:X4(Q ) andhM ( x 3,2:4) = 5Xl,X2(Q )- F or any biquadratic polynomial h ( x i , x 2) E J*2 se^: r i ( x i ) = ^x2(h ) and r 2( x 2) = 6Xl(h ). Then (6.80)

ik {h 12) = ik{hu ),

(6.81)

gk( r 2) = g k ( n ) ,

k = 2 ,3, k = 2,3.

In other words, in the diagram (6.78), the pairs of biquadratic polynomials on the opposite edges have the same invariants i 2,is, and all fou r quartic polynomials ri have the same invariants g2,gs-

These results suggest the following approach to the classification of mul­ tiaffine equations Q = 0 modulo Mobius transformations. Suppose that, for a given Q E 4 , the four quartic polynomials r i ( x i ) associated to the ver­ tices of the quadrilateral in the diagram (6.78) are known. Then one can use Mobius transformations to bring these polynomials into a canonical form. After that, one can reconstruct the edge biquadratics hli from the pairs of vertex polynomials r*,rj. Finally, one can reconstruct the multiaffine Q from the edge biquadratics.

6.11.3. Synthesis: ascending from quartic r to biquadratic h . Ac­ cording to formulas (6.71), (6.76),

SXl(SXj,Xk( M[ Q} ) ) = A 2A 2A 2M[SXl(SXj,Xk(Q))} = C A r 2M[n],

248

6. Consistency as Integrability

where C = A fA ^ A ^ A 2. Since the polynomial Q is defined up to an arbitrary constant factor, we may assume that Mobius changes of variables in the equation Q = 0 induce transformations Ti

A ~2M[ri]

of the polynomials r T h i s allows us to bring each r* into one of the following six canonical forms: (6.82) r — (x 2 — 1) ( k 2x 2 — 1), r = x 2 — 1 , r = x 2, r = x, r = 1 , r = 0 , according to the six possibilities for the root distribution of r: four simple roots, two simple roots and one double, two pairs of double roots, one simple root and one triple, one quadruple root, or, finally, r vanishes identically. Note that in the first canonical form it is always assumed that k2 ^ 0,1, so that the second and third forms are not considered as particular cases of the first form. Not every pair of such polynomials is admissible as a pair of polynomi­ als at two adjacent vertices, since the relative invariants of the polynomials of such a pair must coincide according to (6.81). We identify all admissi­ ble pairs, and then solve the problem of reconstruction of the biquadratic polynomial (6.79) by the pair of its discriminants (6.83)

5x (h) =

Sy(h) - h2 - 2hhyy = n ( x ) ,

h2 -

2hhxx =

r2{y),

which is equivalent to a system of 10 (bilinear) equations for 9 unknown coefficients of the polynomial h.

Lemma 6.25. (Reconstructing biquadratic from two discriminants) Nondegenerate biquadratic polynomials with a given pair of discriminants

(r i(x ), 7*2 ( 2/)) the canonical fo rm (6.82) exist i f and only i f n ( x ) = r (x ) and 7*2 ( 2/) — r (y ) one and the same canonical fo rm r. These polynomials h can be brought into the following normal form s, possibly after Mobius transformations of x ,y not affecting r :

(q4)

r (x ) = (x 2 —l)(A:2x 2 —1) :

h = -^~(x2 + y2 —2Axy —a2 —k2a2x 2y2), 2 a

where A 2 = r(a);

(q3)

r(x) =

5 —x2 :

h=

.1

zsin (aj

(x2 + y2 - 2 cos(a) xy) - S S1^ a ) , 2

where 5 = 0 , 1 ; (q )

2

r{x) = x :

h = - ^ ( x - y)2 - ^ { x + y) +

(q l)

r(x) = 5 :

h = ^

- —---- where 2a 2

(5 = 0 , 1 .

;

6.11. Proof of the classification theorem

249

In the cases (q4), (q3)<$=i and (q2) any biquadratic h with a given pair of discriminants (r(x ), r ( y ) ) is automatically of the form given in the lemma; in the cases (q3)s=o and (q l) an additional Mobius transformation might be necessary to bring h to this form (for instance, in the case (q l)^ o ? that is, r ( x ) — r ( y ) = 0 , any biquadratic h = (nxy + Ax + fiy + v ) 2 has this pair of discriminants, and any Mobius transformation of x , y preserves this form of h). One clearly sees the origin of the elliptic curve in the case (q4): the solution of the problem of finding a biquadratic h ( x , y) with the pair of discriminants ( r ( x ) , r ( y ) ) in the case r ( x ) = ( x 2 — 1) ( k 2x 2 — 1) is parametrized by a point (a, A ) of the corresponding elliptic curve. Introducing the uniformizing variable a by a — sn(a), so that A — sn'(a) = cn(a)dn(a), we can write the corresponding biquadratic (q4) in the form

One can recognize this polynomial as the addition theorem for the elliptic function sn(x;fc); more precisely, h ( x , y ; a ) = 0 if and only if x = sn(£;fc) and y = sn(?7; k) with £ — rj = ± a .

6.11.4. Synthesis: ascending from biquadratics h ^ to multiaffine Q. The next step is the reconstruction of the multiaffine polynomials from the biquadratic ones. In doing this, the following facts are useful (they are proven by a direct computation).

Lemma 6.26. (Reconstructing multiaffine equation from edge bi­ quadratics) F or any multiaffine polynomial Q G J 4 , with the notation = SXk,Xl ('Q ) G 7%, the following identities hold:

where t = h f ^ h 34 -

(6 .86 )

+ h 23h™X:i

2Q X1

h ^ h 34 - h}x\h22, + h 23 h l* - h 233 h 34

Q

h 12 h 34 - h u h 23

Identity (6.85) shows that h 11 can be expressed through the other three biquadratic polynomials (provided i z (h 12) -f 0). Differentiating ( 6 .86 ) with respect to x 2 or X4 leads to a relation of the form Q 2 = F [ h 12, /z23, /i34, fe14], where F is a rational expression in terms of hli and their derivatives. There­ fore, if the biquadratic polynomials on three edges (out of four) are known, then Q can be found explicitly. Of course, it is seen from Lemma 6.26

250

6. Consistency as Integrability

that not every set of three biquadratic polynomials comes as hli from some Q e ? l

P r o o f o f T h e o re m 6.18. We demonstrate the reconstruction procedure in the most interesting case (Q4). Let the polynomials h 12, /i23, h 34 and h u be of the form (q4), with parameters denoted by (a, A ), (b ,B ), (cl, A ) and (b ,B ), respectively, all of them lying on the elliptic curve A 2 = r(a ). The relative invariants of h 12 and hM must coincide because of (6.80), and it is easy to check that this condition allows only the following possible values for (a, A ): (a, A ),

( —a , —A ),

-j—^ ( a , —A ),

-j—^ ( —a ,A ).

According to (6.71), a Mobius change of variables in the equation Q — 0 yields SXk,Xl( M [ Q } ) = A kA iM [6 Xk^ , (Q )] = C A - l A ~ l M \ h %

where C = A 1A 2A 3A 4. Since Q is only defined up to a multiplicative constant, we may assume that a Mobius change of variables induces trans­ formations hlj 1 ^ A r ’ A ~ l M [ h 13}

of the biquadratic polynomials

. I 11 particular, if

hu = h (x s , x 4 ] - a , - A )

or

hM = / i^ 3 ,x 4;

- ^ 2 )?

then the corresponding Mobius transformation, X3 1—> —X3 or X3 1—> l/(fcx3), will change h 34 to - h ( - x 3, X i ; - a , - A ) ,

resp.

-

k x % h {^ -^ x i \

- ^ ),

both of which coincide with h ( x 3 , £4 ; a, A ) due to the symmetries of the polynomial (q4). Thus, performing a suitable Mobius transformation of the variable x 3 (which does not affect the polynomial r ( x 3)), we may assume without loss of generality that (a, A ) = (a, A ). After that, the polynomial h u is uniquely found according to formula (6.85), and it turns out that the equality (b , B ) = ( b , B ) is fulfilled automatically. Thus, the change of one variable allows us to achieve the equality of the parameters corresponding to the opposite edges of the square. A direct computation using formula ( 6 .86 ) yields the equation a (x\x 2 + X3X4 ) + b(x\x± + X2 X3 ) - c ( x { x 3 + x 2 x 4 ) - abc(l + k 2x i x 2x 3x 4 ) = 0 ,

where c = (a B + b A )/ (l —k 2a2 b2). The uniformizing substitution a = sn(a), b = —sn(/?), so that A = sn'(a), B = sn'(/3), and therefore c = sn(a — /?), brings it to the form (Q4).

251

6.11. Proof of the classification theorem

Also in the other cases (Q 1)-(Q 3), suitable Mobius changes of the vari­ ables X2 , £ 3 , £4 allow us to bring the polynomials into the form h 12 = h ( x i , x 2 ',ct), h 23 — h ( x 2 , £ 3 ; /?), hM — h ( x 3,x ^ o t ). A direct computation with formula (6.85) proves that this yields h lA = /i(xi, £4 ; /3). Then the multiaffine Q is found by using ( 6 .86 ). □

6.11.5. Putting equations Q = 0 on the cube. Proof of Theorem 6.19. Given a 3D consistent system (6.68) with all equations of type Q, one can use the Mobius transformations from (M o b ) 8 to bring all six equations into the canonical form from the list Q. Since by Theorem 6.22 biquadratics coming to an edge from two adjacent faces must coincide up to a constant factor, all six equations have to be of one and the same type (Q 1)-(Q 4). Moreover, the parameters k 2 in the case (Q4) and 6 in the cases (Q3), (Q l) have to be the same on each face of the cube. There­ fore, the equations on all faces may differ only by the values of a and fi. Consider the equations corresponding to three faces adjacent to one vertex, say to x: A ( x , x 1 ,X 12 ,X2) = Q { x , x i , x i 2 , x 2;a,(3) = 0, B ( x , x 2, X23, x3) = Q {x, x2, X23, 23 ; /?, 7) = 0,

C (x ,x 3,x 13, Xi) = Q (x ,x 3, x i 3, x 1; 7 , a ) = 0. We will show that one can write these three equations as (6.87)

Q( x , x i , x i j , x f , a i i , aj ) = 0.

For the polynomials (Q 1 )-(Q 4) from the list Q we have: (6.88)

h 12(x ! , x 2)

=

6X3tX4Q (x i,X 2 ,X 3 ,X 4 ;a ,0 ) = K (a ,f3 )h { x i,x 2 ;a ),

(6.89)

h 14(x i , x 4)

=

6X2,X3Q ( x i , x 2, x 3, x 4i;a,f3) = K((3,a)h(xi,x4\f3),

with the biquadratics h ( x , y ; a ) listed in Theorem 6.25 as (q l)-(q 4 ). Thus, we find: A 0,1(x ,x i) = K (a ,(3 )h (x ,x i; a ),

A ° '2( x , x 2 ) = K,(j3,a)h(x,x 2 \P),

B ° '2(x, x 2) =

B ° ’3(x, x3) =

k (/3,

7 )h (x , x 2; (3),

C 0,3(x ,x 3) = k ( 7 , a )h (x , x 2; 7 ),

k(j

, 0 )h (x , x3; 7 ),

C 0,1 (x, x i) = K,(a,‘y ) h ( x , x i ; a ) .

According to the second statement of Theorem 6.22 and to formula (6.74), the following relations must hold: (6.90) h ( x , x i; d )

— --------- r = m ( a , Q ) ,

h (x ,x i;a )

/*m \

(6-91)

h ( x , x 2-,0)

,a

— ------- — =m(/3,/?),

h { x , x 2;(J)

h { x , x 3; j )

—---------r = 171( 7 , 7 ). h ( x , x 3; 7 )

/c(a,^)K(/3,7)«(7,a) . 5 m (a, a)m(/?, 0 )m { 7 , 7 ) = -1« (^ , a )K ( 7 ,/ ?)«(a , 7 )

252

6. Consistency as Integrability

For the most complicated case (Q4), to which we will restrict ourselves in this proof, the biquadratic (q4) is given in (6.84), and a direct computation gives ft(a,/?) = 2sn(a) sn(/?) sn(a — (3). Equations (6.90) yield that a may only take the values ± a , which correspond to m (a ,d ) = ± 1 , and analogously for /?,7 . Equation (6.91) with the above-mentioned values of «(a,/3) yields m (a, ct)m(/3, (3)m ( 7 , 7 ) = 1. Thus, up to a change of enumeration, two cases are possible: a — a,

(3 = (3,

or

7 = 7

a — a,

f3 = —/?,

7

=

—7.

In the first case the equations A = 0, B = 0, C = 0 have the desired form (6.87) with a\ = a, a 2 = (3, 0:3 = 7 . In the second case it is enough to observe that the equation B = 0 is not affected by the replacement of parameters (/?, 7 ) with ( —/?, —7 ) = ( —/?, 7 ), which again leads to the desired form (6.87) with a\ = a, a 2 = —/3, <23 = 7 . Continuing to argue in a similar manner for faces adjacent to other vertices, one shows that the signs of edge parameters can always be adjusted on the whole cube as in system (6.72).

□

6.12. Classification of discrete integrable 2D systems with fields on vertices. II In the previous two sections, we classified quad-equations Q = 0 of type Q, that is, those with all nondegenerate edge biquadratics, and showed their 3D consistency. However, quad-equations of type H, i.e., those with (some of) the edge biquadratics being degenerate, are by no means less interesting or less important. It is enough to mention that the very prominent Hirota equation is of type H (which is the reason for the choice of the latter no­ tation). A classification of multiaffine equations of type H seems to be a rather complicated and tiresome task. Nevertheless, postulating some addi­ tional properties, a classification can be achieved. Our assumptions for the quad-equation Q = 0 will be as follows: ► Linearity. The left-hand side of the equation (6.92)

Q ( x i , x 2 , x 3 , x 4;a,(3) = 0

is a polynomial of degree 1 in each variable, depending on two parameters assigned to the edges. ► (6.93) (6.94)

Symmetry. The function Q has the symmetry properties Q ( x i , x 2,x3,x4',a,l3)

=

eQ(x\, £4 ,^ 3 , x 2](3, a),

e = ±1

aQ( x2, X3, X4, xi ] f 3, a ) ,

a = ±1.

► Tetrahedron property. The value X123, existing due to 3D consistency, depends on x\, x 2 and £ 3 , but not on x.

253

6.12. Classification of discrete integrable 2D systems. I I

The symmetry properties are natural to require, because to enable us to pose our equations on arbitrary quad-graphs, the equations should not depend on the enumeration of vertices. Note that the normal forms of the list Q possess these symmetries. The tetrahedron property is admittedly a less natural classification assumption, but it holds for the vast majority of known interesting examples, including all the equations of the list Q and the Hirota equation itself; see formula (6.32). We consider here the problem of 3D consistency for equation (6.92) in the sense of the system (6.72), with one and the same polynomial Q. Due to the symmetry assumption, the natural transformation group, which can be used to put the equation in the normal form, is essentially smaller than in Section 6.10; namely, all vertex fields should be acted on by one and the same Mobius transformation.

Theorem 6.27. (Classification of symmetric equations with tetra­ hedron property) Any 3D consistent quad-graph equation (6.92) possess­ ing the linearity, symmetry, and tetrahedron properties is equivalent, modulo Mobius transformations acting simultaneously on all variables

and mod­

ulo point transformations of the parameters a, (3, to one of the equations of the following lists. List Q from Theorem 6.18:

(Q4)

sn(a)(xi£2 + ^3^4) -sn (/?)(xix4 + x2^3) -s n (a -y 3 )(x ix 3 + x2^4) + sn(a — (3) sn(a) sn(/?)(l + k 2 x\X2 XzX4) = 0,

(Q3)

s m ( a ) ( x i x 2 + x s x 4 ) - s m ( p ) ( x i x 4 + x 2 x s ) - s m ( a - / 3 ) ( x i x s + x 2 x 4 [)

+5 sin (a — (3) sin(a) sin(/?) = 0, (Q2)

a (x \ x 2 + £ 3^ 4) - (3{x 1 X4 + £ 2^ 3) — (a — (3)(x 1X3 + X2X4 ) + a ( 3 ( a - / 3 ) ( x i + X 2 + X 3 + X 4i) - o i { 3 ( a - l 3 ) ( a 2 - a f 3 + ( 3 2) = 0,

(Q l)

a(xix2 +

£

3^ 4)

-

/? ( £

1£4 + x 2x 3) - ( a - P) (x \xs

+ £

2£ 4 )

+5a/3(a — (3) = 0 ; list H:

(H3)

a(£i£2 + £ 3^ 4) - (3(xiX4 + £ 2^ 3) + 8 ( a 2 - (32) = 0,

(H2)

(£1 - £3) (£2 - X4 ) + {(3 - a )(x \ + £2 + £3 + X4 ) + (32 - a 2 = 0,

(H I)

(£1 - £3) (£2 - £4) +

P

~ ol =

0;

and list A :

(A2)

s in (a )(£ i £4 + £ 2^ 3) ~ sin(/?)(£i£2 + £ 3^ 4) — S in (a — /?)(1 + £ i £ 2# 3£ 4) = 0 ,

(A I )

a (£ i£ 2 + £ 3^ 4 ) ~ 0 ( x 1 X4 + £ 2^ 3) + (a - P){x\Xs + X2X4 ) —5a(3(a — (3) = 0.

Remarks. 1) The parameter S in equations (Q3), (Q l), (H3), (A I ) can be scaled away, so one can assume without loss of generality that 5 = 0 or 6 = 1.

254

6. Consistency as Integrability

2) If one extends the transformation group of equations by allowing Mobius transformations to act on the variables £ i,£ 3 differently than on £ 2, #4 (white and black subgraphs of a bipartite quad-graph), then equation (A2) turns into (Q3)<5=o by the change (£ 2,^ 4) (l/#2, l/#4), and equa­ tion (A I ) turns into (Q I) by the change (# 2, £ 4) ►{ ~ x 2, —X4 ). So, really independent equations are given by the lists Q and H. 3) Equation (H3) is the most general in the list H, since (H I) and (H2) can be considered as its limiting cases. Note that (H I) is the discrete K dV equation and (H3)«j=o is a version of the Hirota equation with the symmetry properties (6.93), (6.94). The general scheme of the proof of Theorem 6.27 is the same as in Section 6.11. We start with the “analysis” part. Due to the symmetry assumption, all edge biquadratics for the polynomial Q(x\, x 2, x 3j X4 ; a, (3) are given by one and the same biquadratic polynomial g(x, y\ a, (3), so that h12(x i , x 2) = 6X3,X4{ Q ) = g ( x i , x 2-,a,P), h u (x i , x 4) = 8X2,X3( Q ) = g { x i , x 4;f3,a).

Moreover, the polynomial g is symmetric: g (x , y ;a ,j3 ) = g (y ,x\ a , (3). L em m a 6.28. (D escen din g from m ultiaffine Q to qu artic r )

The

biquadratic g(x,y\ot,[3) admits a representation

(6.95)

g (x, y; a, (3) = fc(a, (3)h(x, y; a),

where the factor k is antisymmetric, k(f3,a) = —k(a,(3), and the coefficients of the polynomial h ( x , y ; a ) depend on a single parameter a in such a way that the discriminant r ( x ) = Sy(h ) does not depend on a at all.

P ro o f. In the proof of Theorem 6.22 we used the previously demonstrated tetrahedron property to derive formula (6.74). In the present setup the tetrahedron property has been postulated, thus we can still use formula (6.74), which, due to the symmetry assumptions, takes the following form: g (x, x i ; a i , a 2 )g (x , x 2; a 2, a 3)p(x, £3; <*3, ai)

= - g { x , £ 1; ai, a 3 )g{x, x 2; a 2, ol\)g(x, £ 3 ; a 3, a 2). This relation implies that the fraction g (x, x i; ct\, a 2 ) /g (x, x\] a\, 0*3 ) does not depend on £ 1, and due to the symmetry it does not depend on x either. We see that the symmetry assumptions has been used in this argument to replace the nondegeneracy of biquadratics which has been required in Theorem 6.22 to come to the same conclusion. We find: g (£ ,£ 1; Ql,Q:2) = g ( x , x i ; a i , a 3)

2)

/c(ai,a3) ’

255

6.12. Classification of discrete integrable 2D systems. I I

where the function

k

satisfies the equation

K ( a i , a 2)K {a 2, a 3)K ( a s , a i ) = - n ( a 2, ai)/c(a3, a 2)K,(ai, a^)-

This equation is equivalent to «(/ ? ,«) = - ^ 7^ « (a ,/ ? ), that is, the function k(a,(3) = (a)/s(a, /?) is antisymmetric. We have: g (x ,y ;a , l3 ) = g ( x , y ; a , 7 )

«(«,/ ? )

K (a,7 )

g(x,y;a,/3) = g ( x , y ; a , 7 )

&(«>/?)

fc(a,7)

’

which implies (6.95). To prove the last statement of the lemma, we notice that h l2(x i , x 2) = fc(a,/3)/i(xi,X 2;a ),

/i14(x i, x 4 ) = —fc(a, 0 )/i(xi, X4 ; 0 ),

and so due to Lemma 6.23, 5X2 (/i(xi, X2; a )) = 5X4 (/i(xi, £4 ;/?)). Thus, r does not depend on a. □ Now we can turn to the “synthesis” part of the proof to make the way back from r to Q. First, we use Mobius transformations to put the poly­ nomial r (x ) into one of the six canonical forms, and look for symmetric biquadratics admitting r (x ) as discriminant.

Lemma 6.29. (Reconstructing symmetric biquadratic from its dis­ criminant) F or a given quartic polynomial r (x ) in one of the canonical forms (6.82), the symmetric biquadratic polynomials h ( x , y ) having r ( x ) as their discriminants are exhausted by the biquadratics ( q l ) - ( q 4 ) from Lemma

6.25 and the following three families: (h3)

r(x ) = x2 :

h = j o x 2y2 + j i x y + 72 ,

7 i “ 47072 = 1;

(h2 )

r(x ) = 1 :

h = 7 0(x + y )2 + 'yi(x + y) + 72 ,

(h i)

r(x ) = 0 :

h = (jo x y + 71 (a; + y) + 7 2) 2.

7?-47072 = 1 ;

Thus, for each of the polynomials r ( x ) = ( x 2 —l ) ( k 2x 2 — 1), r ( x ) = l —x 2 and r(x ) = x there exists only one family of symmetric biquadratics with discriminant r(x ), given by (q4), (q3)<$=i, resp. (q2). On the contrary, for the polynomials r (x ) = x 2, r (x ) = 1 and r (x ) = 0 , in addition to (q3)^=o and (q l), we have the branches (h3), (h2), (h i).

Proof of Theorem 6.27. Having found the biquadratics /i, one can finally reconstruct the multiaffine Q with the edge biquadratics satisfying the con­ ditions of Lemma 6.28, which is achieved by solving linear systems for the coefficients of Q. The conditions of Lemma 6.28 are necessary for 3D consis­ tency with the tetrahedron property. It turns out that they are also almost sufficient.

256

6. Consistency as Integrability

More precisely, the biquadratics (q l)-(q 4 ) uniquely determine functions Q satisfying the conditions of Lemma 6.28. These are functions (Q 1)-(Q 4),

respectively, and they are 3D consistent with the tetrahedron property. It the cases (h l)-(h 3 ) a careful analysis reveals several Mobius non­ equivalent families of biquadratics which allow for multiaffine functions Q satisfying the conditions of Lemma 6.28: Oi h = — ^

(h3)

1 — cr

/o o

( xy

^n

+ 1)

1

oi2

1 —a 2

xy,

h = xy + a, h = xy;

(h 2 )

» =£<*+»>’ -§. h = x + y + a;

(h i)

h= ^(x+^2 ’ h = l.

For all families, with the exception of (h3)<5=o and (h i) which contain no pa­ rameters at all, the resulting functions Q are (A 2 ), (H3)«$= i, (H 2 ), ( A 1)<5= i, and (A 1) (5=o, which turn out to be 3D consistent with the tetrahedron prop­ erty. Thus, in all these cases the necessary conditions of Lemma 6.28 turn out to be also sufficient. In each of the remaining two cases (h3)<$=o and (h i), there is a family of polynomials Q, containing an arbitrary skew-symmetric function /c(a,/3) = —fc(/3, a), which satisfies the conditions of Lemma 6.28: Q

=

(1 + fc(a, P ) ) ( x 1X2 + X3 X4 ) - (1 - fc(a, 0 ) ) ( x 1X4 + x 2x 3),

Q

=

(x i ~ x 3 ) ( x 2 - x 4 ) + f c ( a , / 3),

respectively. These are the only two cases when not all equations passing the necessary test of Lemma 6.28 turn out to be 3D consistent. The first of these candidate equations is 3D consistent if and only if 1 + k(a,/3)

a

1 — k(a, j3)

(3 ’

up to a point transformation of parameters. This is equation (H3)<s=o- The second candidate equation is 3D consistent if and only if k(a,(3) = (3 — a, up to a point transformation of parameters. This is equation (H I). □

6.13. Integrable discrete Laplace type equations The geometric construction of the double 2) for a given surface graph 9, described in Section 6.5, leads to a construction of a collection of integrable

257

6.13. Integrable discrete Laplace type equations

discrete Laplace type equations, based on a deep and somewhat mysterious property of 3D consistent quad-graph equations with fields on vertices. We will write the quad-equation (6.36) on a bipartite quad-graph D in slightly modified notation as (6.96)

Q ( x 0, 2/1, x i , y2; a i, a 2) = 0 ;

see Figure 6.20. For notational simplicity, vertices x stand here for the corresponding fields f ( x ) ; the edges ( xq , y \), (xo, y2) carry the labels a i, a 2j respectively. V2

Vi F ig u re

6.20. A

labelled

bipartite

face

of

F ig u re

a

6 .2 1 . Three-

leg form of a quad-

quad-graph;

fields on vertices.

equation.

Definition 6.30. (Three-leg form) A n equation (6.96) possesses a threeleg fo rm centered at the vertex xo i f it is equivalent to the equation

(6.97)

ip(x0,y i;a ti) - ip(x0,y 2; a 2) = <j)(x0, x i ; a i , a 2)

with some functions 0 , (j). The terms on the left-hand side correspond to the “short” legs

(

x q

, y i), ( a ^

to the “long” leg

(

x q

,

x i

y2)

) G

G

E(T>), while the right-hand side corresponds

E ( S).

Summation of quad-graph equations for the flower of quadrilaterals ad­ jacent to the “black” vertex xq G V (S ) (see Figure 6.5) immediately leads, due to the telescoping effect, to the following statement.

Theorem 6.31. (From quad-equations to Laplace type equations) a)

Suppose that equation (6.96) on a bipartite quad-graph D possesses a

three-leg form.

Then the restriction of any solution f : V(T>) —> C to the

“black” vertices V ( $ ) satisfies the discrete Laplace type equations,

(6.98)

^ ^ Xk£

star(xo )

(x0 )

Q'k'i &k-f-l)

258

6. Consistency as Integrability

b) Conversely, given a solution f : V ( S ) —► C of the Laplace type equations (6.98) on a simply connected surface graph S> there exists a oneparameter family of extensions f : Vr(2)) —> C satisfying equation (6.96) on the double D. Such an extension is uniquely determined by the value at one arbitrary vertex o f V ( S*).

Sometimes it is more convenient to write the three-leg equation (6.97) in the multiplicative form: y\\ ai)/\I>(xo?V2 \<^2) =

(6.99)

with some functions become multiplicative:

£*2)

x\;

so that the Laplace type equations (6.98) also

(6.100)

$ ( x 0, x k; a k, a k+1 ) = 1. XfcG star(xo)

It turns out that all 3D consistent equations of the lists Q, H, and A from Theorem 6.27 admit three-leg forms. They are presented in the following theorem for the lists Q and H (the results for the list A follow from these results). T h e o re m 6.32. (T h re e -le g form s o f in tegrab le qu ad-equ ation s) The three-leg forms f o r all equations of the lists Q , H from, Theorem 6.27 read as follows:

(Q4): Multiplicative three-leg form with <2>(x\ y ; a, p ) = ^ (x , y: a — p ), ( 6 .101 )

^ o ^ i;«) = 0 o ?(Ao Y0 + a )r Sn! v° + a )! -~ m —a sn(Ao —a s n!(Av i)r

where x = s n (X ) , and & ( X ) is the Jacobi theta-function. (Q3)s=\: Multiplicative three-leg form with <S>(x,y;a, /3) = ty(x,y;ot —/3), ~r~/ x sinfXn + a ) — sin (X i) * x 0 , xi ; a = — ^ , sm(Ao — a ) — sm (A i)

6.102

where x = sin(X ).

(Q3)(5=o; Multiplicative three-leg form with $(x,j/;a,/3) = ^ (x , y\a/p), (6.103)

® (x 0, xi; a) =

— — . xo — ax\

(Q 2 ): Multiplicative three-leg form with <3>(x\ y\ a, (6-104) where x = X

« ( x 0, x i ; o ) = |y ° + (A 0 - a )2 - A f

2

P)

= ^ (x , y ;a — p ),

259

6.13. Integrable discrete Laplace type equations

y;

a,

< j > ( x , y \ a , ( 3)

=

( Q l ) s = i : Multiplicative three-leg fo rm with

inc\ (6.105) (a

tfif

\

Xq + a -

(3 )

=

ty(x,y;a

—( 3),

Xi

w (x o ,x i;a ) = -------------- . Xq — a

( Q 1 ) s=

q:

Additive three-leg fo rm with

(6.106)

^ (x q , x i;

a)

— X\

^(x,y\ a

— (3 ),

= — -— . xo -

X\

(H 3 ); Multiplicative three-leg fo rm with (6.107)

§(x,y\OL,(3) = — — ^

a x — fjy

,

\&(£0 ,£ i;a ) = £o£i + 6a.

(H 2 ): Multiplicative three-leg fo rm with (6.108)

$ (x ,y ,a ,(3 ) = - — V + a ^ x - y - a + (3

^ (x o , xy, a ) = x 0 + £i + a.

(H I): Additive three-leg fo rm with (6.109)

4>{x, y; a,

0) =

a — (3 -------,

x -y

tp(x0, x i ; a )

=

x 0 + x\.

P r o o f. The proof of this theorem is obtained by a direct computation; see Exercise 6.15. However, this does not give any insight in how these three-leg forms could be found. A general way to derive three-leg forms is the subject of Exercise 6.16. □ R em ark . It should be mentioned that the existence of a three-leg form allows us to derive (and, in some sense, to explain) the tetrahedron property of Section 6.11. Indeed, consider three faces adjacent to the vertex £123 in Figure 6.19, namely the quadrilaterals (xi, £ 12, £ 123, £ 13), (£ 2 , £ 23, £ 123, £ 12)* and (£ 3 , £ 13, £ 123, £ 23)* A summation (resp. multiplication) of the three-leg forms (centered at £ 123) of equations corresponding to these three faces leads to the additive equation (6.110)

^ ) ( x i 23, X i ; a 2, a 3) + 4>(x 1 2 3 ,^ 2 ; « 3 , <*i)

+


0,

respectively to the multiplicative equation ( 6 .111 )

$(£123, £ 1; « 2 , ^3)$(£i23, £25 <^3 , Q i)$(£i235£35

OL2 ) = 1 -

This is the equation which relates the fields at the vertices of the “white” tetrahedron in Figure 6.19. Note that it can be interpreted as a discrete Laplace type equation coming from a spatial flower with three petals. The functions ( j ) ( x , y \ a , f3) (resp. $(£, y; a , /?)) corresponding to the “long” legs, yield additive (resp. multiplicative) Laplace type equations on arbitrary planar graphs. Studying the list of Theorem 6.32, one sees that there are only six “long” legs functions. Three of them are rational in £, y;

260

6. Consistency as Integrability

each of the corresponding Laplace type systems admits two different exten­ sions to a quad-graph system: one from the list Q, where the form of the “short” legs coincides with the form of the “long” ones, (Q3)<$=o, (Q1)<5=i, and (Q1)<5=o? and the other from the list H, with different “short” legs, (H3), (H 2 ), and (H I), respectively. The other three functions <E> are rational in y only, and require a uniformizing change of the variable x. The correspond­ ing Laplace type systems admit only one extension to a quad-graph system, (Q4), (Q3)*=1, and (Q 2 ).

Theorem 6.33. (Integrability of Laplace type equations) Additive Laplace type equations (6.98) with the ulong legs” functions 0(x, y\a, (3), resp. multiplicative Laplace type equations (6.100) with the “long legs” func­ tions <£(x, y; a , /?) from Theorem 6.32, are integrable in the sense of Section

6.4.

Proof. Observe that the functions 0, resp. $, always contain the parame­ ters a, (3 in the combination a — (3. This means that the edge parameters of the extension of the Laplace type system to a quad-graph system are only defined up to an additive constant A. Let the equations of the extended quad-graph system read Q ( x Qi Vk, Xfc, Vk -\-1 i C^k

^5 &k-\-1

^)

Rewrite this equation as a Mobius transformation, ( 6 . 112 )

Vk+l

?A) [yk] •

L'i.XQi Xk, OLh,

Then the (normalized, if necessary) matrix L ( x o, a^, A) is the tran­ sition matrix across the edge (x$,xk) € E ( S), or along the edge (yk,Vk+i) £ E ( S*), in the zero curvature representation of the Laplace type system on

□

Example. Consider the additive Laplace type system on 9, corresponding to the equation (Q I )<$=() (cross-ratio equation):

(6.113)

£

a t ~_at+1 = 0.

XfcG star(xo)

^

^

The extension to the quad-equation reads: (

xq

(yk

-V k )(x k -

- 2 / a h - i) =

Xk)(yk+ 1

“

Xq)

Qfc + A a

^+1

+ A ’

which can be written as the Mobius transformation ( 6 .112 ) with the matrix

261

6.14. Fields on edges: Yang-Baxter maps

These matrices give a zero curvature representation of the Laplace type system (6.113) on an arbitrary surface graph 9.

6.14. Fields on edges: Y an g-B ax ter maps We now turn to the study of another large class of 2D systems on quadgraphs with fields assigned to the edges. In this situation it is natural to assume that each elementary quadrilateral carries a map F : X x I - > X x X , with X being the set where the fields x ,y take values, so that F ( x , y ) — (x 2,y i); see Figure 6 .22 . The concept of 3D consistency of such maps may be interpreted in several ways, depending on the initial value problem one would like to pose on the elementary 3D cube.

y

x

F ig u re

6 .2 2 . M ap encoded by an elementary quadrilateral with fields

on edges.

One way is to choose the initial data x , y , z on three edges of an elemen­ tary cube adjacent to one vertex. One computes first F ( x , y ) = ( x 2, y i ) ,

F ( y , z ) = (yz,z2),

F ( z , x ) = (* 1, 0:3 ),

and then F { x z,yz) = (x2z,yi3),

F ( y i , z j ) = (y i3 ,z r2),

F { z 2, x 2) = {zi2, x 23),

so that there are two a priori different answers for any of the fields X23 , 2/13, 212 with two indices; see Figure 6.23. D efin itio n 6.34. (3 D consistent m ap ) A map F : X x X - + X x X i s called 3D consistent i f the two answers f o r each of the fields (# 23, 2/13*212) in Figure 6.23 coincide f o r any initial data ( x , y , z ) . An important example of a 3D consistent map, coming from discrete differential geometry, is discussed in Exercises 6.19, 6.20. Definition 6.34 has one notational inconvenience: since each initial edge is used on the first step by two different maps, it is not possible to express the property of 3D consistency in terms of compositions of maps. This can be overcome by a different choice of initial data, namely by choosing them on a path consisting of three edges of three different coordinate directions.

262

6. Consistency as Integrability

£23

y / /A\ \

£3

1 11

1

Zi

1

\

1 1 1

V -

1H 110

221

/ //

£

F ig u r e 6.23. 3D consistency of 2D systems with fields on edges.

This leads to the notion of Yang-Baxter maps (traditionally denoted by R rather than by F ) . D efin ition 6.35. (Y a n g -B a x te r m ap ) A map i ? : X x X —> X x X is called a Yang-Baxter map i f it satisfies the Yang-Baxter relation

(6.115)

R\2

o

i? i3 O i?23 =

#23 ° #13 0 #12,

where each Rij : X 3 t—> X 3 acts as the map R on the factors i , j of the Cartesian product X3, and acts identically on the third factor.

Equation (6.115) is understood as follows. The fields x , y are supposed to be assigned to the edges parallel to the 1-st and the 2-nd coordinate axes, respectively. Additionally, consider the fields z assigned to the edges parallel to the 3-rd coordinate axis. Initial data are the fields £, y , z on a path consisting of three edges of different coordinate directions; see Figure 6.24. The left-hand side of this figure corresponds to the composition of maps on the left-hand side of equation (6.115), which are visualized as maps along the three front faces of the cube: R

23 ( y,

z) = (2/3, 22),

3

R l ( x , Z2) = ( x 3, ZU ),

Rn(x

3 , 2/3)

= (^ 23, V u ) -

(Here and below we slightly abuse the notation by omitting the arguments on which our maps act identically.) Similarly, the right-hand side of the figure corresponds to the chain of maps on the right-hand side of (6.115), which are visualized as maps along the three back faces of the cube: R u ( x , y ) = (£ 2 , 2/1 ),

R i z ( x 2 , z) = (x23,*i),

# 23 ( 3/1 , 2 1 ) = ( 2/1 3 , 2 12 )-

So, equation (6.115) assures that the two ways of obtaining (£23,2/13,^12) from the initial data (£, 2/, z) lead to the same results.

263

6.14. Fields on edges: Yang-Baxter maps

^23

#23

F i g u r e 6.24. Yang-B axter relation.

The notion of the zero curvature representation makes perfect sense for Yang-Baxter maps, and can be expressed as (6.116)

L ( x , A)L(y, A) = L { y u A)L(.x2, A).

There is a construction of zero curvature representations for Yang-Baxter maps with no more input information than the maps themselves, close in spirit to Theorem 6.4. Consider a parameter-dependent Yang-Baxter map R(a. , /3), with parameters a, fi G C assigned to the same edges of the quadri­ lateral in Figure 6.22 as the fields x,y, opposite edges carrying the same parameters. Although this can be considered as a particular case of the general notion, by introducing X — X x C and R ( x , a ; y , fi) = i?(a, /?)(#, y), it is convenient for us to keep the parameter separately. Thus, in Figure 6.24 all edges parallel to the x (resp. y, z) axis carry the parameter a (resp. fi, 7 ), and the corresponding version of the Yang-Baxter relation reads: (6.117)

R n ( a , {3)Rvi(a, 'y)R 2s(P, 7 ) = #23(/?, 7 )-Ri3 ( « , 7 )-Ri2 (<*, /?)•

T h e o re m 6.36. (Z e ro curvature rep resen tation for Y a n g -B a x te r m aps) Suppose that there is an effective action of the linear group G = GL(iV, C) on the set X (i.e., A G G acts identically on X only if A — I ) , and that the Yang-Baxter map R ( a , p ) has the following special form :

(6.118)

x 2 = B(y,/3,a)[x],

yi = A {x , a, fl)[y}.

Here A, B : X x C x C —> G are some matrix-valued functions on X depending on parameters a and fi, and A[x] denotes the action of A G G on x G X. Then , whenever (# 2 ,y i) —

/?)(#? y)> we have

(6.119)

A (x ,a ,\ )A (y ,l3 ,\ )

— A (y i, fi, \)A(x< 2 , a, A),

(6.120)

£ (y ,/ 3 ,A )S (x ,a ,A )

=

S ( x 2, a , A)B(yi,/3, A).

264

6. Consistency as Integrability

In other words, both A (x , a, A) and B curvature representations f o r R.

x(x, a, A) (o r B T (x, a, A) ) fo rm zero

P ro o f. Look at the values of 212 produced by the two sides of the YangBaxter relation (6.117): the left-hand side gives z u = A (x , a, 7 )A (y , f3,7 ) [z], while the right-hand side gives z \2 — A(y\, /?, ^ ) A ( x 2 ->a, 7 ) [z]. Now since we assume that the action of G is effective, we immediately arrive at the relation A (x , a , - y )A (y , P , j ) = A (y i, 0 , ^ ) A ( x 2 ,a , - f ) ,

which holds whenever (x 2,yi ) = R (a , (3 )(x,y ). This coincides with (6.119), an arbitrary parameter 7 playing the role of the spectral parameter A. Similarly, one could look at the values of X23 produced by the two sides of (6.117): the left-hand side gives X23 = B(y$, (3 , a ) B ( z 2 ,'y,ct)[x\, while the right-hand side gives X23 — B (z , 7 , a )B (y , /?, a)[x]. Effectiveness of the action of G again implies: B (y 3, (3, a ) B ( z 2, 7 , a ) = B (z , 7 , a ) B ( y , 0, a ),

whenever ( 2/3, 22) = 7 )(y >z )- This coincides with (6.120); here the role of the spectral parameter A is played by an arbitrary parameter a. □ In order to cover all known examples, the scheme of Theorem 6.36 must be extended in the following way. We say that A (x , a, A) gives a projective zero curvature representation for the Yang-Baxter map R if the relation (6.116) holds up to multiplication by a scalar matrix c/, where c may depend on all the variables in the relation. Assume that the action of G = G L ( N , C) on X is projective, i.e., scalar matrices and only they act trivially. Then the previous considerations show that the matrices A (x , a, A) and B ~ l ( x , a, A) give projective zero curvature representations for the corresponding YangBaxter maps (6.118). In practice, the natural choices of matrices A , B in (6.118) actually lead to proper zero curvature representations, as the following examples show. E xam p le 1 : A d le r m ap. Here X = C P 1 and the map has the form /*ioi\ ( 6 .121 )

a ~-— $ , x~ = y -----x + y

y~ = x

@ ~ a x+y

Then _

(i-a x2 + x y - ( ( 3 - a ) y = x ------ -— = -------— -----------= A (x ,a ,p )\ y ), x+y x+y

where m

w

(

A (x ,a ,X ) = I

x 1

x2 + a - \

x

and the group G = GL(2, C) acts projectively on C P 1 by Mobius transfor­ mations. In this example B ( x , a , A) = A (x , a, A), so the matrices B T = A T provide us with an alternative zero curvature representation.

265

6.14. Fields on edges: Yang-Baxter maps

Example 2: Interaction of matrix solitons. Our next example comes from mathematical physics. The matrix Korteweg-de Vries equation Ut + 3U U X + 3UXU + Uxxx — 0 admits one-soliton solutions of the form U (x, t) = 2 a 2Psech2(a x — 4a 3t), where a is the parameter measuring the soliton ve­ locity, and the matrix amplitude P must be a projector: P 2 = P . Projectors of rank 1 have the form P = ^rjT / (£, 77). It turns out that the change of the matrix amplitudes P of two solitons with velocities a\ and after their interaction is described by the following Yang-Baxter map: R ( a 1 , 012) : (£i,f7i; €2 , 7/2 ) 1001

c

(£i,?h;£ 2 ,% ),

, 2 a 2 (£i, >72) + 7----------T77----- rs 2> (a i - a 2) ( f 2, rj2)

t

(6.122)

6 = 6

(6.123)

£2 = £2 + 7----------777----- r fl,

t

,

,

2 a 2 { 6 ,^i) „ ( « i - « 2) ( « , % )

V i = V i + 7----------w 7----- \V2 ,

2 a i (^2 , 771)

.

2 a i(^ i,% )

V 2 = m + -/----------T77-----\^?1•

( a 2 — a i ) ( £ i , 771)

( a 2 - <*i)(£i, 771)

In this example X is the set of projectors P of rank 1 which is the variety Qp7V-i x QpW-i^ ancj a projective action of the group G — G L ( N , C) on X is induced by A [(£, 77)] = ( A ^ A r j ) . It is easy to see that formulas (6.123) can be written as (£ 2 ,% ) =

>t(£l, ^ 7 l , « l , ^ 2 ) [ ( ^ 2 , % ) ]

with the matrices 4 /^

%\

r

2a

£t7T ( f , 7/)

77, a , A) — / + ---------- • — — - .

A- a

Thus, the matrices A(£, 77, a, A) give a projective zero curvature representa­ tion for the interaction map, but it is not difficult to see that this is actually a genuine zero curvature representation. As in Example 1 , i?(£, 7/, a, A) = ^(£, Vf

^) •

Example 3: Yang-Baxter maps arising from geometric crystals. Let X = Cn, and define f i : X x X - > X x X by the formulas (6.124)

P

xj = Xj —

$j = yj

P

j = 1 ,..., n,

v j -1 where n

( 6 .125 )

Pj =

/ a—1

( a=l

n \k= 1

\

n

xj+k

n /c = a+ l

/

(in this formula subscripts j + k are taken (mod n)). Clearly, the map (6.124) keeps the subsets X Q x c X x X, where n

Xq — { ( x i , . . . , xn) G X .

x k — a }, k= 1

266

6. Consistency as Integrability

invariant. It can be shown that the restriction of R to X a x Xp may be written in the form (6.118). For this, the following trick is used. Embed X a x X (3 into CP " " 1 x C P " " 1 via J ( x , y ) = ( z ( x ) , w ( y )), where z (x ) = (1 : zi : • • • : zn- 1),

w(y) = (wi : • • • : wn-1 : 1 ),

j

zi =

n

n xk ,

Wj =

k —1

n

yk ■

k —j + 1

Then it is easy to see that in the coordinates (z, w) the map R is written as z = B ( y , /?, a ) [ z ] ,

w = A (x , a, 0)[w\ ,

with certain matrices 5 , A from G — G L(n ,C ), where the standard projec­ tive action of GL(n, C) on CP n_1 is used. Moreover, a simple calculation shows that the inverse matrices are cyclic two-diagonal:

(6.126)

(3, a )

yi

-1

0

0

?/2

—1

... ...

0

o o

0

0

0

y3

...

0

0

o

0

0

ljn- i

-1

0

0

... ...

\ —a (

(6.127)

A ~ (x, a, 0 )

=

X\

0

x2

0

... ...

0

—1

xs

...

0

0

0

0

0

0 0

0 ...

0 0

0

...

x n- i -1

\

yn ) -0

\

0

0 0

xn J

To be more precise, the matrices A , B are defined only up to multiplication by scalar matrices. These scalar matrices are chosen in (6.126), (6.127) in such a way that the dependence of the matrices B ~l , A ~ l on their “own” parameters (/?, resp. a ) drops out, so that the only parameter remaining in the zero curvature representation is the spectral one. In other words, the zero curvature representation does not depend on the subset XQ x Xp to which we restricted the map. It can be checked that this is actually a genuine (not only projective) zero curvature representation. Note also that in this example the matrices B T coincide with A (so they cannot be used to produce an alternative zero curvature representation for R ).

6.15. Classification of Y an g-B axter maps Consider Yang-Baxter maps i ? : l x l - > l x l , {x ,y ) ►( u , v ) in the following special framework. Suppose that X is an irreducible algebraic variety, and R is a birational automorphism of X x X. Thus, the birational

267

6.15. Classification of Yang-Baxter maps

map i ?-1 : X x X —> X x X, {u ,v ) i—> ( x , y ) is defined. This is depicted in the left square in Figure 6.25. Furthermore, a nondegeneracy condition is imposed on R: rational maps u (-,y ) : X —> X and v(x, •) : X —> X must be well defined for generic x, resp. y. In other words, birational maps H r X x X ^ X x X , (x, v) i—> (u, y) and .R-1 : X x X —> X x X, (u ,y ) i—► (x,t>), called companion maps to i?, must be defined. This requirement is visualized in the right square in Figure 6.25. Birational maps R satisfying this condition are called quadrirational. A formal definition of a slightly more general notion (where different spaces are allowed for the arguments x and y) looks as follows.

Definition 6.37. (Quadrirational map) Let X i , X 2 be two irreducible algebraic varieties over C. A rational map F : X i x X 2 —> X\ x X 2 , identified with its graph, an algebraic variety T p C Xi x X 2 x Xi x X 2 , is called quadrirational i f f o r any fixed pair ( x , y ) G Xi x X 2, except possibly some closed subvarieties of codim,ension > 1, the variety T p intersects each of the sets { # } x { y } x Xi x X 2, X\ x X 2 x { x } x { y }, Xi x { y } x { x } x X 2 , and

{ x } x X 2 x Xi x { y } exactly once, i.e., if T p is a graph of fou r rational maps F, F ~ \ F , F ' 1 : Xi x X 2 ^ Xi x X2.

u

u

F i g u r e 6.25. A map F on X x X, its inverse and its companions.

It is possible to classify all quadrirational maps in the case Xi = X 2 — C P 1; we give a short presentation of the corresponding results. Birational isomorphisms of C P 1 x C P 1 are necessarily of the form ( a i oo\

'

z?.

‘

.. _

a (y )x ^ + ub{y) \y> c (y )x + d(y) '

_

A (x )y + B {x ) C (x )y + D (x ) ’

where a ( y ) , . . . , d(y) are polynomials in y, and A ( x ) , . .., D ( x ) are polyno­ mials in x. For quadrirational maps, the degrees of all these polynomials are < 2. Depending on the highest degree of the coefficients of each fraction in (6.128), we say that the map is [1:1], [1:2], [2:1], or [2:2]. The richest and most interesting subclass is [2:2]. For the maps of this subclass the polyno­ mials A ( x ) = A ( x ) D ( x ) — B ( x ) C ( x ) and S(y) = a(y)d (y ) — b(y)c(y) are of

268

6. Consistency as Integrability

degree four. A quartic polynomial belongs to one of the following five types, depending on the distribution of its roots: I: four simple roots, II: two simple and one double root, III: two double roots, IV: one simple and one triple root, V: one quadruple root. It turns out that a necessary condition for a map of the subclass [2:2] to be quadrirational is that the polynomials A (x ) and 6(y) are simultaneously of one of the types I-V . Sufficient conditions are more complicated and can be expressed in terms of singularities of the map F, i.e., those points ( M e C P 1 x C P 1 where both the numerator and the denominator of at least one of the fractions in (6.128) vanish: (6.129)

a(v)£ + K v ) — 0,

c(r/)£ + d(rj) — 0

or (6.130)

A ( 0 v + B ( 0 = 0,

C ( 0 v + D (£ ) = 0.

For instance, if both polynomials A ( x ) and S(y) are of type I, then the nec­ essary and sufficient condition for the quadrirationality of the map (6.128) is that the roots Xi,yi (i — 1 , . . . , 4) of A (x ) , 5 ( y ) can be ordered so that both equations (6.129), (6.130) be satisfied for (£,??) = (x ;,y i), i = 1,... ,4; in other words, the four singularities of both fractions in (6.128) be at the points ( x u y i ). One can find normal forms for all quadrirational [2 :2] maps with respect to the action of the natural transformation group, which in this case is the group (M ob ) 4 of Mobius transformations acting independently on each of the fields x, y, u, v.

Theorem 6.38. (Classification of quadrirational maps on C P 1 x( Any quadrirational [2:2] map on C P 1 x C P 1 is equivalent, under some change of variables acting by Mobius transformations on each field x, y, u , v inde­ pendently , to exactly one of the following five normal forms: V F\:

„

Fu :

_

Fm :

P u — ayP, y ^ a [3

a D v = pxr,

D

P —

{ 1 - 0 )x + 0 - a + { a - l ) y (3(1 - a ) x + ( a - (3)yx + a(f3 - 1) y ’

x _ „ ax — dy + (3 — a v = - P , P = ----------------------, x —y

y _x _ _ ax — (3y u=^P, v = - P , P = -------a jj

x —y

6.15. Classification of Yang-Baxter maps

Fry:

Fy:

u = yP,

u = y + P,

v = xP,

269

P = 1 + —— —,

?; = x + P,

x -y

P = - — x -y

with suitable constants a,j3.

Each one of the maps F i , . . . , F y is an involution and coincides with its companion maps, so that all four arrows in Figure 6.25 are described by the same formulas. Note also that these maps come with the intrinsically built-in parameters a,/3. Neither their existence nor a concrete dependence on parameters is presupposed in Theorem 6.38. A geometric interpretation of these parameters can be given in terms of singularities of the map; it turns out that the parameter a is naturally assigned to the edges x, u , while P is naturally assigned to the edges y,v. For instance, for the map F\ the parameter a is nothing but the cross-ratio of the four roots X{ of the polynomial A (x ), and similarly /3 is the cross-ratio of the four roots yi of the polynomial S(y). The most remarkable fact about the maps F j , ... ,F y is their 3D con­ sistency. For T = I, II, III, IV or V, denote the corresponding map F j of Theorem 6.38 by Fj(a,/3), indicating the parameters explicitly. Moreover, for any OLi,OL2,Oi3 G C, denote by F ^ = F<j(cti,aj) the corresponding maps acting nontrivially on the z-th and the j-th factors of (C P 1)3.

Theorem 6.39. (Norm al forms of quadrirational maps on C P 1 x C P 1 are 3D consistent) F o r any 7 = I, II, III, IV or V, the system of maps F^ is 3D consistent, and also satisfies the Yang-Baxter relation with parameters

(6.117).

Proof. The proof can be obtained by a direct computation (Exercise 6.22). It will also follow from Theorem 6.40 below, after we provide a geometric interpretation of the maps F j. □ Actually, 3D consistency of quadrirational maps on C P 1 x C P 1 holds not only for the normal forms F j but also under much more general circum­ stances. The only condition for quadrirational [2:2] maps consists in match­ ing singularities along all edges of the cube. Similar statements hold also for quadrirational [1 :1] and [1 :2] maps, so that in the case Xi = = CP1 the properties of being quadrirational and of being 3D consistent are related very closely. The maps F j of Theorem 6.38 admit a very nice geometric interpreta­ tion. Consider a pair of nondegenerate conics Q i, Q 2 on the plane CP2, so that both Q i are irreducible algebraic varieties isomorphic to C P1. Take

270

6. Consistency as Integrability

X G Q i, Y E Q 2 ? and let £ = ( X Y ) be the line through X , Y (well-defined if X 7£ Y ) . Generically, the line £ intersects Q\ at one further point U ^ X , and intersects Q 2 at one further point V i=- Y . This defines the map

(6.131)

7 : Qi x Q 2 -

Q x x Q2,

J (X , Y ) = (£/, V );

see Figure 6.26 for the R 2 picture. The map 3 is quadrirational, it is an involution and moreover coincides with both its companions. This follows immediately from the fact that, knowing one root of a quadratic equation, the second is a rational function of the input data. Intersection points X G Q i H Q 2 correspond to the singular points (X , X ) of the map 9\

Figure 6.26. A quadrirational map on a pair of conics.

Generically, two conics intersect at four points; however, degeneracies can happen. There are five possible types I —V of intersection of two conics: I: II: III: IV: V:

four simple intersection points; two simple intersection points and one point of tangency; two points of tangency; one simple intersection point and one second order tangency point; one point of third order tangency.

All conics sharing a quadruple of points build a linear family, or a pencil of conics. There are five types I- V of pencils of conics. Using rational parametrizations of the conics: C P 1 3 x y-* X ( x ) E Ql

C

C P2,

resp.

C P 1 9 y ^ Y ( y ) € Q 2 C C P2,

it is easy to see that 7 pulls back to the map F : (x , y ) (u , v ) which is quadrirational on C P 1 x C P 1. One shows by a direct computation that the maps F for the above five situations are exactly the five maps listed in Theorem 6.38. Now, we obtain the following geometric interpretation of the statement of Theorem 6.39.

271

6.15. Classification of Yang-Baxter maps

F i g u r e 6.27. 3D consistency on a linear pencil of conics.

Theorem 6.40. (3D consistent maps on a pencil of conics) Let Qi, i — 1, 2, 3, be three nondegenerate members of a linear pencil of conics. Let X G Q i , Y E Q 2 and Z G Qs be arbitrary points on these conics. Define the maps $ij as in (6.131); corresponding to the pair of conics ( Q i , Q j ). Set ( X 2, Y 1) = GF12( X , Y ) , ( X ^ Z , ) = ? i s ( X , Z ) , and (Y3, Z 2) = ? 23( Y , Z ) . Then

x 23 = ( X 3y3) (6.132)

n

( X 2Z 2)

e

Qu

y 13 = ( X 3y3)

n{Y

^ )

e

Q 2,

z 12 = {Y1Z 1) n ( X 2Z 2) e Q 3.

In other words, the maps

are 3D consistent.

Proof. We will work with equations of lines and conics on CP 2 in homo­ geneous coordinates, and use the same notation for geometric objects and homogeneous polynomials vanishing on these objects. Construct the lines a = ( Y Z ), b — { X Z ) and c = (-^ ^ ), respectively. Let X 2= (cnQi)\X,

Y 1 = ( c n Q 2) \ Y ,

Xs = ( b n Q i ) \ X ,

Y3 = ( a n Q 2) \ Y .

Next, construct the line C — (X 3Y3), and let x 23 = ( C n Q i) \ x 3,

Yis = { c n q 2) \ y 3.

Finally, construct the lines A = (Y iY i3) and B = (X 2X 23). We have four points X , X 2, X 3 and X 23 on the conic Q i, and two pairs of lines (C, c) and (J5, b) through two pairs of these points each. Therefore, there exists /ii G C P 1 such that the conic Q\ has the equation Q i = 0 with Q i — fJ'iBb + Cc.

272

6. Consistency as Integrability

Similarly, the conic Q 2 has the equation Q 2 = 0 with Q ‘2 — 1^2 Aa + Cc.

Consider the conic

Qi ~ Q 2 = MiBb - fi2 Aa = 0. It belongs to the linear pencil of conics spanned by Q i and Q 2 - Furthermore, the point Z = a fl b lies on this conic. Therefore, the conic Q\ — Q 2 must coincide with Q%, which has therefore the equation Q% = 0 with Q 3 = fi\Bb - fi2Aa. Moreover, the two points Z 2 = a D B and Z\ = b fl A also lie on Q 3 . Since Z 2 G B , we have B = { X 2 Z 2 ). Similarly, since Z\ G A , we have A = {Y\Z\). Finally, we find that the point Z 12 = A fl B = (Y \ Z i) fl ( X 2 Z 2 ) also lies on Qz, which is equivalent to (6.132). □

6.16. Discrete integrable 3D systems The major part of this chapter has been devoted to the very rich theory of integrability of 2D equations, the root of which has been identified in their 3D consistency. In this last section we turn our attention to integrability of 3D systems, now understood as 4D consistency. The most striking feature is that the number of integrable systems drops dramatically with the growth of dimension: we know of only half a dozen of discrete 3D systems with the property of 4D consistency. All of them are of a geometric origin and in fact appeared already in Chapters 2, 3, and 4. We are going to briefly discuss their general algebraic features. In the 3D context, there are a priori many kinds of systems, according to where the fields are assigned: to the vertices, to the edges, or to the elementary squares of the cubic lattice. 6.16.1. F ield s on 2 -faces. Consider first the situation when the fields (assumed to take values in some space X) are assigned to the the elementary squares. Denote by a, b, c the fields attached to the 2-faces parallel to the coordinate planes 12, 13, 23, respectively. The system under consideration is a map F : X 3 i-» X3, which we write as F (a, 6 , c) = ( 73a, 726, t \ c ) — (fl3,&2,ci). One can think of the fields a, 6, c as sitting on the bottom, front, and left faces of a cube, and as, 62, c\, on the top, back, and right faces. This is visualized in Figure 6.28. The concept of 4D consistency of such a map assumes that one can extend it to a four-dimensional square lattice. Thus, in addition to the fields a, b, c, there are fields d , e, f , attached to the 2-faces parallel to the coordinate planes 14, 24, 34, respectively. Initial data a, b, c, d, e, f are the

273

6.16. Discrete integrable 3D systems

Figure 6.28. 3D system on an elementary cube: a map with fields on 2-faces.

fields on six 2-faces of a 4D cube adjacent to one vertex. They allow one to apply the map F on four 3-faces of a 4D cube (the inner, bottom, front, and left ones in Figure 6.29):

Here

F 123 : ( a ,b ,c )

(a3,62,c i),

F X2a '■ ( a ,d ,e )

(a4,d2,e 1),

F 134 : ( b, d, /)

( 64, ds, /i),

F 234 : (c, e, f )

( 04, e3, f 2).

denotes the map F acting on a 3-face of the coordinate directions ‘B i j k - Now one can apply the map F on the other four 3-faces (the outer, top, back, and right ones): F

F ijk

123 • (a 4 , 64, 04) 1-* (a34,&24,ci4),

F 134 • ( 62^ 25/2)

(&24?rf23»/l2),

F n 4 : (a3,^3,e3) i-> (a34,d23,ei3),

^234 • ( c i ,e i ,/ i) i-> (C14, ei 3 , f n ) •

Thus, one obtains two answers for each of the six fields a34, 624> ci 4 ? c?23, ^13, /12, and the map F is 4D consistent if these pairs of answers identically coincide for all six fields and for all initial data. We mention here two examples of systems of the kind just discussed, both of geometric origin. The first is the discrete Darboux system which describes Q-nets in the affine setting; see Section 2.1.3. For this system, each 2-face of the coordinate direction 25^, i < j , carries a field consisting of a pair of real numbers ( 7 ^, 7 ^). The map is given by the formulas (6.133)

j.

Tk li j =

1

Ijk lk j

Theorem 6.41. (4D consistency of the discrete Darboux system) The discrete Darboux system (6.133) is 4D consistent.

The second example is the star-triangle map which describes T-nets; see Section 2.3.8. For this system, each 2-face of the coordinate direction 25^, i < j , carries just one real number a^, and the convention = —aji holds.

274

6. Consistency as Integrability

F i g u r e 6.29. Initial data and results of two-stage application of a 4D consistent map with fields on 2-faces.

The map is given by the formulas (6.134)

Tka,ij = ---------- •— -----------------, ttijtijk + ^jkaki + (lkiaij

A symmetric appearance of this formula one would like to consider a\j with i < minus signs in the denominator. Thus, fields a = a\2 1 b — a i 3, c = a2:h used at star-triangle map is written as (6.135)

k t -L i , j .

is due to the above convention. If j only, there would appear some in the index-free notation for the the beginning of this section, the

^--------, [ -------F{a, b, c) = (a3, b2, c \) = ( - ■ a------ , \ab + be — ca ab + b e — ca ab + be — ca

Theorem 6.42. (4D consistency of the star-triangle map) The startriangle map (6.135) is

consistent.

As in the case of 2D systems (see Section 6.14), our definition of consis­ tency cannot be written in terms of composition of maps, since each piece of the initial data is used simultaneously in two different maps. It turns out to be possible to change the initial value problem on a 4D cube in such a way that the resulting consistency condition can be formulated in terms of compositions. It is not difficult to realize that for this the initial data should be prescribed on six 2-faces (of all six two-dimensional coordinate directions) which form a surface topologically equivalent to a disk. Such a surface is depicted on the left in Figure 6.30. One can apply to this initial surface two different sequences of flips of the kind depicted in Figure 6.28, both leading to the surface on the right in Figure 6.30. One sequence starts with flipping the inner 3-face, and then

275

6.16. Discrete integrable 3D systems

Figure 6.30. Initial data surface for a map satisfying the functional tetrahedron equation, and the result of its four-fold flipping.

the top, front, and right ones. Denote the maps corresponding to these flips by Sijk] they are “companion maps” for the original F, i.e., they arise from F by regarding it along various diagonals of the basic cube. There appears a composition of maps: Sl23 : (a’ b, c) *5l34 • (^2 ,^ 2 ,/ )

(tt3 , b‘2 , Cl), ( 624, ^23, f l ) ,

Si24 : (<*3 , d, e) I—> (a 34, C?2 , ei), S 234 • ( c i , e i, / i) H-> (ci 4 , ei 3 , /12).

Another sequence starts with flipping the left 3-face, and proceeds with the back, bottom, and outer ones: S234 • (c, e, /) *—►(C4 , e3 , f 2), S i24 : (a, CZ3 , e3 )

( 04 ,^ 23, 613),

S 134 : ( 6 , d, f 2)

( 64, cfo, /12),

S 123 : ( 04 , 64,^4 ) i-> ( 034, 624,^ 14).

The requirement that the two chains of maps lead to identical results can be thus encoded in the formula (6.136)

S 234 0 S i 34 0 S 124 0 S i 23 = S i 23 0 S i 24 0 S 134 O S234.

D efin itio n 6.43. (F u n ction al tetra h ed ron equ ation ) A X 3 is said to satisfy the functional tetrahedron equation if where each S ^ is a map on X 6(a ,b ,c,d ,e , f ) acting as S of the Cartesian product X 6 corresponding to the variables faces parallel to the planes i j , i k , jk , and acting trivially on

map S : X 3 ^

(6.136) holds, on the factors sitting on the the other three

factors.

Thus, we see that the concept of functional tetrahedron equation es­ sentially coincides with the concept of 4D consistency of 3D systems with fields on 2-faces, the main difference lying in how the initial value problem is posed for the system at hand. It can be demonstrated (see Exercise 6.26)

276

6. Consistency as Integrability

that the 4D consistency of the star-triangle map (6.135) is translated into the following result.

Theorem 6.44. (Star-triangle solution of the functional tetrahe­ dron equation) The map (6.137)

S(a, b, c) = (a3, b2, ci) = ( — —^ —— — , a + c - abc, — ——— — ^ \a + c — abc a + c — abc)

satisfies the functional tetrahedron equation (6.136).

The map (6.137) is related to the map (6.135) via conjugation by b •—> 1/6. One of the integrability features of 4D consistent maps (or, equivalently, of maps satisfying the functional tetrahedron equation) is a 3D analog of the zero curvature representation. For the map (6.137) it is discussed in Exercise 6.27.

6.16.2. Fields on vertices. Another version of 3D systems deals with fields assigned to vertices. In this case each elementary cube carries just one equation (6.138)

Q(x, xi, x2, x3, X12, X23 , Z i 3 , £ 123) = 0,

relating the fields x E X in its eight vertices. Such an equation should be solvable for any of its eight arguments in terms of the other seven. This is shown in Figure 6.31.

F i g u r e 6.31. 3D system on an elementary cube: an equation with fields on vertices.

The 4D consistency of such a system is defined as follows. Initial data on a 4D cube are 11 fields x, X* (1 < i < 4), x^ (1 < i < j < 4). This data allow one to uniquely determine, by virtue of (6.138), all the fields Xijk ( l < 2 < j < / c < 4 ) . Then one has fo u r different possibilities to find X1234, corresponding to the four 3-faces adjacent to the vertex X1234 of the 4D cube; see Figure 6.32. If all four values coincide for any initial data, then

277

6.16. Discrete integrable 3D systems

equation (6.138) is 4D consistent. For such systems, one can consistently impose equations (6.138) on all three-dimensional cubes of the lattice Z4.

2-1234

Z34

^124

F i g u r e 6.32. 4D consistency of a 3D system with fields on vertices.

The only examples of 4D consistent equations with one scalar field at­ tached to each vertex we know are related to the star-triangle relation and appear through different factorizations of the face fields . Given a (complex-valued) solution (6 139)

of equation (6.134), the relations

Tkaij _ Ti ajk _ rj aki &ij djk &ki

yield the existence of a function z : Z m —►C such that ZiZj

(6.140)

—

zz.ij

i < j.

It is essentially unique (up to the values on coordinate axes whose influence is a trivial scaling of z). Due to (6.134), the function z satisfies on any 3-face the following equation: (6.141)

Z^ijk

^i^jk “t- ZjZik

^k^ij — 0 ?

i

j

k.

We will call it bilinear cube equation (other names used in the literature are discrete B K P equation and Hirota-Miwa equation).

Theorem 6.45. (4D consistency of the bilinear cube equation) Equa­ tion (6.141) is 4D consistent. The value 2:1234 is given by

(6.142)

££1234 - £12^34 + £13*24 “ £23*34 = 0 .

278

6. Consistency as Integrability

It should be noted that (6.142) essentially reproduces the bilinear cube equation itself. Moreover, 2:1234 does not actually depend on the values %i, 1 < i < 4. It relates the fields on the even sublattice only, and this can be considered as an analog of the tetrahedron property of Section 6.11. Sometimes the bilinear cube equation (6.141) is written in a slightly different and more symmetric form, with only plus signs on the left-hand side. On every three-dimensional subspace these two forms are easily transformed into one another. However, this cannot be done on the whole of Z m, if m > 4. Equation (6.141) with only plus signs on the left-hand side is not 4D consistent and cannot be posed on Z m. Another scalar cube-equation closely related to the star-triangle map can be derived, if one remembers that the latter appears as the compatibility con­ dition for the Moutard equations (2.51) which govern T-nets. Considering Moutard equations with scalar fields, we see that, given a (complex-valued) solution aij of (6.134), the relations (6.143)

mj = — — - ,

X j — Xi

i ± j,

define a function x : Z m —►C. It is determined uniquely as soon as its values on the coordinate axes are prescribed. Now relations (6.139) yield equalities for x:

(6 1 4 4 )

= ( Xi -

Xj)(xk -

Xij k)

(x -

x i j ) ( x ik -

xjk)

Actually, for a given triple of indices, all equations (6.144) for various per­ mutations of indices are equivalent, so that there is only one such equation for a 3-face. It is natural to call this the double cross-ratio equation. This is an equation of the type (6.138), uniquely solvable for a field x at an ar­ bitrary vertex of any 3-face, provided the fields at the other seven vertices are known.

Theorem 6.46. (4D consistency of the double cross-ratio equation) Equation (6.144) is J^D consistent.

A remarkable feature of equation (6.144) is its high symmetry grade: actually, it admits a symmetry group D$ of the cube. More precisely, if one writes (6.144) in the form (6.138) with a multiaffine polynomial Q, then for any reflection S from D& we have Q o S = —Q. It turns out that this property alone already characterizes the double cross-ratio equation.

Theorem 6.47. (Symmetry characterization of the double cross­ ratio equation) Consider a multiaffine polynomial Q ( x , x\, x 2, £3, £12, £23, # 13, £123) such that f o r any reflection S from the symmetry group of the cube

D$, Q o S — asQ with as G {1, —1}. Then:

279

6.17. Exercises

a) Equation Q = 0 either coincides with (6.144), or is Mobius-equivalent to one of the linearizable equations ( X + X [ 2 + x 23 + x i s ) i

(# 1 + x 2 + x 3 + #1 2 3) — 0?

XX12X23X 13 =b X 1 X2 X3 .7*123 = 0,

.X*.X‘lX 2X3.Ti2X23;r 13X123 = ±1. b) I f a$ = —1 f o r any reflection S in D$. then equation Q = 0 coincides with (6.144).

6.17. Exercises 6 . 1 . Verify by a direct computation that the complex Hirota and cross-ratio

equations are 3D consistent: check equations (6.34), (6.32). 6 .2 . Check that (6.34) can be put in the form of the cross-ratio equation:

(6.145)

9(/i,/3,/2,/i23) =

OL2 ~~ O13

6.3. Verify by a direct computation the 3D consistency of the discrete KdV equation (/ - /12X /2 - f i ) = 0 - a.

6.4. Consider a 2D integrable system on Z 2 with a zero curvature represen­ tation with transition matrices L(e, A), under periodic boundary conditions f m ri = fm+M,n+N- Consider a monotone path ( c i , . .., ep) with the starting point (rn, n) and the end point (rn + M , n + N ) (so that p = M + N ) . Define the monodromy matrix as T mjl = L (z p, A) • • •L(t\, A). Show that (i) for a given initial point (m, n) the matrix Tm?ri does not depend on the choice of the path; (ii) the eigenvalues of Tm?n do not depend on the initial point (m,n).

6.5. Compare solutions of the Cauchy problem for the linear wave equation fl2 - f l ~ f 2 + f = 0 011 the regular square lattice without defects and with a localized defect, as

in Figure 6 .12 . Conclude that even weak defects are nontransparent for the linear wave equation. Remark: Theorem 6.7 does not hold for the linear wave equation, since the latter does not contain parameters, and therefore does not admit a zero curvature representation with a spectral parameter. 6 .6 . Describe the quad-surface

in Z 3 canonically corresponding to the dual kagome lattice: characterize vertices and edges of Q<j).

6.7. Provide details in the proof of Theorem 6.14 on 3D consistency of the noncommutative discrete K dV equation.

280

6. Consistency as Integrability

6 .8 . Prove 3D consistency of the following equations with fields / in an

associative algebra A: <*(fi - fv 2 + /?)(/ - f l - a ) - 1 = (3(f2 - f 12 + a ) ( f - f 2 - f3)~l

(no-commutative version of equation (Ql)<$=i), and (1 - ot2) ( f i - 0 f i 2) ( a f - / i ) - 1 = (1 - /?2) ( / 2 - a f 12) ( P f - /2 ) - 1

(noncommutative version of equation (Q3)^=o)- Hint: Try to proceed as in the proof of Theorem 4.26 given in Section 4.3.7, or in the proof of Theorem 6.13. 6.9. Prove that for any Q G 7\ the polynomial P defined by the following formula also belongs to 7\: Q

(

Q

x2

Qx4

It is called the accompanying polynomial for

Qx 1

Qx 3

Q X1X2

Q x 2x 3

Qxjx4

Q

x -JX4

\ •

j

Q.

6 . 1 0 .* Find the accompanying polynomial P defined in (6.146) for the poly­ nomials (Q 1)-(Q 4).

6.11. Show that the necessary and sufficient condition for the equation (/ — /12X /1 — /2) + k(a, (3) = 0 with a skew-symmetric fc(a,/?) = —fc(/?, a ) to be 3D consistent is that k ( a , (3) = fi — a, up to a point transformation of parameters.

6 .12. Consider the quad-equation (.x - 6 j)(x ij - O j)(xi - O i)(xj - ei) - (x - e i){x ij - Oi)(xi - e j ) ( x j - o j) = 0

with a pair of parameters (e*, 0 { ) assigned to the edges of the i-th coordinate direction. Show that this equation is 3D consistent but does not possess the tetrahedron property. Check that its edge biquadratics are degenerate, as they should, according to Theorem 6.22. 6.13. Show that if one drops the condition of nondegeneracy of biquadrat­ ics in Theorem 6.25, then one can find biquadratics with discriminants ri (a:), 7*2( 2/) of different canonical forms. Prove that the list of such bi­ quadratics, modulo Mobius transformations, is given by: r2(y) = y2 :

ri(x) = x 2 - l ,

n ( x ) = x, r2(y) = 1 : ri(x) =

1,

r2(y) =

0:

h = a y 2 ± xy +

4a

;

h = ^(y - a ) 2 - x; h = 7 0y 2 + 71 y + 72,

6.14. Consider the following list of quad-equations:

7 i - 47072 = 1­

281

6.17. Exercises

(H3e)

a (x \ x 2 + X3 X4 ) - (3 {x\x 4 + x 2xz) + {a 2 - (32)

( H 2 e)

(xi -

xz)(x2 -

X4) + ( P -

= 0,

a ) ( x i + x 2 + £3 + X4) + ft2 -

a2

+c(/3 — a ) ( 2 x 2 + a + /?)(2 x 4 + a + /?) + e(/? — a

( H l e)

)3

—

0,

(xi - x 3) ( x 2 - x 4) + {(3 — a ) ( l + ex2x4) = 0.

These equations are e-deformations of the list H. Compute their edge bi­ quadratics and vertex quadric polynomials, and show that they are of the type considered in Exercise 6.13. Check that these equations possess the rhombic symmetries Q ( x i , x 2,x 3 ,x 4 ,a,/3) = —Q (x 3, x 2, xi, x4, /3, a ) = - Q ( x i , x 4 , x 3, x 2,/?, a),

which flip the pairs of vertices of the same color. Show that each of these equations can be put in a 3D consistent fashion on the multidimensional lattice if one respects the bi-coloring of the vertices: Q (x , Xi , X ij , X j , ol%, Oij) — 0 , Q ( x i k , x k,x jk ,x i2 3 ,0 ii,a j) = 0 ,

{i, j,fc } = {1,2,3}.

The tetrahedron property is fulfilled. 6.15. Check the tree-leg forms of quad-equations of the lists Q, H in Theo­ rem 6.32. 6.16.* Prove the following recipe for finding the three-leg form of an ar­ bitrary multiaffine equation Q ( x i , X 2 , x 3 ,X4) = 0 : for any permutation ( i , j , k , l ) of ( 1 , 2 , 3 ,4), (6.147)

f i j ( x i , x j ) + f i k( x i , x k) + f u ( x i , x i ) =

4>( xi) ,

where (6.148)

f fij(xi,Xj)=

j

dx' ..

3—

•■

T,

r i J{X i,X j)

h% 3( x i , x j ) =

5XktXl(Q ).

This is the three-leg form of the equation Q = 0 centered at x*. The function (f>{xi) on the right-hand side of (6.148) can be regarded as resulting from the integration constants in the integrals on the left-hand side. It can be determined by considering some (in fact, any) solution of Q — 0 singular at Xi , for which = 0, hkl = 0, hjl = 0. 6.17.* Use the recipe of Exercise 6.16 to derive the three-leg forms of equa­ tions of the lists Q, H. 6.18.* For equations of the list Q it is possible to derive the 3D consistency from the existence of the three-leg form. Indeed, suppose equation (6.36) admits symmetries (6.93), (6.94) and a three-leg form (multiplicative, for definiteness) ^ f ( x i , x 2-,a)/'i/(x1,x 4 ;p ) = ^ ( x i , x 3; a - (5).

282

6. Consistency as Integrability

Prove that then it is 3 D consistent.

6.19.* Consider the following map with fields from an associative algebra A with unit: (6.149) Prove its 3D consistency in the sense of Definition 6.34. 6 .2 0 . Definition (6.149) of the map F from the previous exercise is equiva­ lently rewritten as

(6.150)

x 2 + y = y\ + x,

x 2y = yix.

Check that this map admits the discrete zero curvature representation L ( x 2, A)L(y, A) = L(yi, \ ) L ( x , A)

with the matrices L ( x , A) — A I + x.

Can you derive this zero curvature representation, following the ideas of Theorems 6.4, 6.36? 6 .2 1 . Specializing the map (6.149), or, equivalently, (6.150) to the case when

all fields belong to the algebra EI of quaternions, show that the real parts of the quaternions x, x 2 are equal, as well as the real parts of the quaternions 2/, ?/i, and that the imaginary parts of the four quaternions build a (nonplanar) quadrilateral in R 3 with opposite sides of equal length (Chebyshev quadrilateral). In other words, there exist a, j3 E R and /, /i, /2, f\2 € = su(2 ) ~ R 3 such that x = a l + ( f l - /),

y = /31 + ( f 2 - /),

x 2 = a l + ( f i 2 - /2 ),

y\ = P I + (/12 - f i ) .

f u - f 1- f 2+ f =

— J — - ( f u - /) x (/ 2 - f i ) 2 \Oi (j )

with

(6.152)

which fixes the proportionality coefficient between the vectors f i 2—f i —f 2+ f and (/12 — /) x ( f 2 — /i); these vectors are parallel due to i f 12 ~ f l ~ f2 + f , f l 2 ~ f ) ~ { f 12 ~ f l ~ /2 + /, /2 - f l ) = 0 ,

which is equivalent to relations (6.151). The 3D consistency of map (6.149) yields the 3D consistency of equation (6.152) with Revalued fields on ver­ tices, if a, (3 is considered as a real-valued edge labelling.

283

6.17. Exercises

6 . 2 2 . Check (by hand or with the help of a computer algebra system) that the maps from Theorem 6.38 satisfy the Yang-Baxter relation.

6.23. Construct zero curvature representations for the Yang-Baxter maps from Theorem 6.38, based on Theorem 6.36. 6.24. Consider a pencil of conics having a triple tangency point at the point (W\ : W 2 : W 3) = (0 : 1 : 0 ) (in homogeneous coordinates on CP2). Conics of this pencil and their rational parametrization are given (in nonhomogeneous coordinates) by the formulas Q(a):

W2 - W 1 2 - a = 0,

X ( x ) = ( W 1( x ) , W 2( x ) ) = ( x , x 2 + a) .

Check that if Q\ = Q( a ) , Q 2 = Q( P ) , then the map £F defined in (6.131) is given in coordinates by Fy of Theorem 6.38.

6.25. Consider a pencil of conics through four points O = (0,0), (0,1), (1,0), (1,1) G C P2, where nonhomogeneous coordinates (W \ ,W 2) on the affine part C 2 of CP 2 are used (any four points on C P2, no three of which lie on a straight line, can be brought into these four by a projective transformation). Conics of this pencil are described by the equation Q(a):

W 2( W 2 - l ) = a W 1( W 1 - l ) .

A rational parametrization of such a conic is given, e.g., by X { x ) = { W l { x ) , W 2{ x ) ) = ( 4 Z ^ , X( 2 ~ Q ) )\xz — a x — ol /

Here the parameter x has the interpretation of the slope of the line ( O X ) . The values of x for the four points of the base locus of the pencil on Q ( a ) are x = a, 00 , 0 and 1 . Show that if Q\ — Q (a ), Q 2 = Q(/?), then the map 7 defined in (6.131) is given in coordinates by F\ of Theorem 6.38.

6.26. The geometric content of the discrete Moutard equation Xij — x = ai j ( x j —Xi) is the parallelism of the lines ( xxi j ) and ( xi xj ) . Therefore, there

are in principle four ways to introduce the field as the proportionality coefficient between the two vectors under consideration: ± a ^ and ± 1 / ai j . Prove that one can introduce the fields for six two-dimensional coordinate directions in Z 4 so that all four maps Sijk in (6.136) be given by the formulas (6.137).

6.27.* Check the following 3D analog of the zero curvature representation for the map (6.137): L 2 3 (c)L is (b )L i2 (a ) = L i 2(a3)Lis(b2) L 2s ( c i ),

where L i 2( a ) =

/ —a 1+ a \

0

1—a

0\

a

0

0

1/

/ —b ,

L 13( 6) =

0

0 1

\l + 6 0

1 — 6\ 0

6 /

,

284

6. Consistency as Integrability

and

0 £23 (c) =

0

-c 1+ c

\

1- cl.

c

J

Can you derive this representation? 6.28. Consider the system of linear equations (6.153)

X2 — x = a{x\ — x) ,

xs — x — b(x\ — x )

for a scalar-valued function x on Z 3. Equations in (6.153), as well as the coefficients a, b, are naturally assigned to triangles; see Figure 6.33. Show

F i g u r e 6.33. Equations on triangles.

that the compatibility of these equations is assured as soon as the coefficients a, b satisfy the following equations: (6.154)

(as - 1)(b - 1) = (b2 - 1)(a - 1),

a3bi = 62ai,

which should be understood as a map ( a , a i , 6, bi) i-> (<23, 62). Valid initial data for such a map can be prescribed on a surface shown in Figure 6.34.

285

6.17. Exercises

Show that, due to the second equation in (6.154), there exists a scalar function / on Z 3 such that a = /2//1 , b = /3//1 , and that this function solves the equation

(6.155)

h~h

+

hzA

J 12

+ .A " A = 0.

J 23

713

The function £ on Z 3 solves the multiratio equation , also known as the Schwarzian discrete K P equation: 1 1

(# 1 j

# 12) ( # 2 -

# 2 3 )(# 3 ~ # 1 3 ) =

_ 1

( X 1 2 - X 2) ( X M - X 3 ) ( X 1 3 - X 1)

'

Give a geometric interpretation of equations (6.153), (6.156) (hint: these equations encode a Menelaus configuration). Can you find a linear system similar to (6.153) which would generate the so-called bilinear octahedron equation (or bilinear H irota equation , or discrete K P equation ): (6.157)

Z1 Z23 + z2zis + z3z12 = 0?

It is natural to call equations of the type (6.155), (6.156), (6.157), which do not involve the fields at the vertices x and #123 of an elementary cube, octahedron equations, as opposed to the general cube equations (6.138). 6.29. Octahedron equations (6.155), (6.156), (6.157) have a sort of 4D con­ sistency property. One imposes such an equation for three 3D coordinate directions (ij4 ): Xl2 =

(6.158)

/(xi,a;2,X4,Xi4,X24)j

X 13 = g (x 1 , x3, x4, # 14, # 34), #23 = h (x 2, # 3 , # 4, # 24, # 34)-

Compared to the usual 4D consistency of cube equations, the vertices x and X ij 4 do not appear in this system, and only three equations are considered. Check that the following holds: equations (6.159)

#123 = f ( g , h, £ 34, g, h) = g (f , h, #24, /, h) = h ( f , g , £ 14, /, g),

are satisfied identically with respect to 11 independent variables (chosen as initial data): # 1 , #2, # 3, #4, #14, #24, #34, #44, #144, #244, #344-

In equations (6.159) the “hat” denotes the shift in the 4-th direction: / = T±(f) = /(£14, £24, £44, #144, #244),

etc.

Verify also that for each of the above systems, an equation of the form (6.160)

k (x i 2 , £ 13, £ 23, # 14, # 24, # 34) = 0

286

6. Consistency as Integrability

holds. For instance, for the multiratio equation (6.156), ( X U ~ X 12) ( X2 4 ~ £ 2 3 )(^ 3 4 ~ £13) _ (£12 -

£ 24) (£23 -

£ 34)(£ 1 3 -

£14)

_ 1 ’

6.18. Bibliographical notes Sections 6.1 , 6 .2 : Continuous and discrete integrable systems. The theory of integrable systems (called also the theory of solitons) is a vast field in mathematical physics with huge literature. The focus of dif­ ferent publications in this area varies from algebraic geometry, enumerative topology, statistical physics, quantum groups and knot theory to ap­ plications in nonlinear optics, hydrodynamics and cosmology. We men­ tion here a selection of mathematical monographs (in chronological order): Toda (1978), Novikov-Manakov-Pitaevskii-Zakharov (1980), Ablowitz-Segur (1981), Calogero-Degasperis (1982), Newell (1985), Faddeev-Takhtajan (1986), Ablowitz-Clarkson (1991), Dubrovin (1991), Matveev-Salle (1991), Hirota (1992), Korepin-Bogoliubov-Izergin (1992), Belokolos-Bobenko-Enol’skii-Its-Matveev (1994), Hitchin-Segal-Ward (1999), Kupershmidt (2000), Rogers-Schief (2002), Babelon-Bernard-Talon (2003), Reyman-SemenovTian-Shansky (2003), Suris (2003), Dubrovin-Krichever-Novikov (2004), Fokas-Its-Kapaev-Novokshenov (2006). Concerning the basic concrete example of these sections, the sine-Gordon equation: the Backlund transformation was found by Backlund (1884); the permutability theorem is due to Bianchi (1892). The zero curvature rep­ resentation is due to Ablowitz-Kaup-Newell-Segur (1973) and Takhtajan (1974). The immersion formula for surfaces with constant negative Gauss­ ian curvature in terms of the frame is in Sym (1985). The discretization (6.12) of the sine-Gordon equation along with its Backlund transformation is due to Hirota (1977b). The geometric meaning was uncovered in BobenkoPinkall (1996a).

Sections 6.3, 6.4, 6.5: Integrable systems on graphs. Our presenta­ tion of the general theory of integrable systems on graphs follows BobenkoSuris (2002a). Examples of integrable systems on the regular triangular lattices were considered in Adler (2000), Bobenko-Hoffmann-Suris (2002) and Bobenko-Hoffmann (2003). The fundamental role of quad-graphs for discrete integrability was understood in Bobenko-Suris (2002a). A different framework for integrable systems on graphs was developed by Novikov with collaborators. In particular, the Laplace transformations on graphs were studied in Dynnikov-Novikov (1997), the theory of discrete Schrodinger operators on graphs was developed in Novikov ( 1999a,b), and the scattering theory on trees is due to Krichever-Novikov (1999).

6.18. Bibliographical notes

287

Sections 6 .6 , 6.7: From 3D consistency to zero curvature repre­ sentations and Backlund transformations. The idea of consistency (or compatibility) is at the core of the theory of integrable systems. It appears already in the very definition of complete integrability of a Hamiltonian flow in the Liouville-Arnold sense, which says that the flow may be included into a complete family of commuting (compatible) Hamiltonian flows; see Arnold (1989). In the discrete context the ( d + l)-dimensional consistency of d-dimensional equations was observed many times. In the case d = 1 it was used as a possible definition of integrability of maps in Veselov (1991). A clear formulation in the case d = 2 was given in Nijhoff-Walker (2001). A decisive step was made in Bobenko-Suris (2002a) and independently in Nijhoff (2002): it was shown that the existence of a zero curvature repre­ sentation follows for two-dimensional systems from the three-dimensional consistency.

Section 6 .8: Geometry of boundary value problems for integrable 2D equations. The discussion of the Cauchy problem on quad-graphs in Subsection 6.8.1 follows Adler-Veselov (2004). Embedding of quad-graphs into cubic lattices as a purely combinatorial problem was studied in a more general setting of arbitrary cubic complexes in Dolbilin-Stan’ko-Shtogrin (1986, 1994) and Shtan’ko-Shtogrin (1992). Theorem 6.8 is due to KenyonSchlenker (2004). The notion of the quasicrystallic rhombic embeddings and the extension to multi-dimensional lattices in Subsection 6.8.2 is due to Bobenko-Mercat-Suris (2005). Note that intersections of f lx> with bricks correspond to combinatorially convex subsets of 2), as defined in Mercat (2004). ^

Section 6.9: 3D consistent equations with noncommutative fields. The notion of 3D consistency in the noncommutative setup was introduced in Bobenko-Suris (2002b), where also the derivation of the zero curvature representation was given. Further examples due to Adler and Sokolov can be found in Adler-Bobenko-Suris (2007). The discrete Calapso equation (6.63) together with its zero curvature representation appeared in Schief (2001). There is a big literature on noncommutative integrable systems. One of the fundamental results in the theory of quantum integrable systems with discrete space-time is the quantization of the Hirota system by FaddeevVolkov (1994). A systematic exposition of noncommutative integrable sys­ tems is given in Kupershmidt (2000). References on discrete noncommuta­ tive systems include Matveev (2000), Nimmo (2006), Schief (2007).

Sections 6 .10 , 6 .1 1 , 6 .12 : Classification of discrete integrable 2D systems with fields on vertices. The classification of discrete inte­ grable 2D systems based on the notion of 3D consistency was given in

288

6. Consistency as Integrability

Adler-Bobenko-Suris (2003, 2007). The first of this papers deals with equa­ tions possessing the cubical symmetry and the tetrahedron property (Theo­ rem 6.27). In Sections 6.10, 6.11 we present the classification of the second paper made under much weaker assumptions (Theorems 6.18, 6.19). Equations (H3)<$=o and (H I) are perhaps the oldest in the lists; they can be found in the work of Hirota (1977a,b). Equations (Q l) and (Q3)$=0 go back to Quispel-Nijhoff-Capel-Van der Linden (1984). Equation (Q4) was found in Adler (1998) (in the Weierstrass normalization of an elliptic curve). This equation in the Jacobi normalization is due to Hietarinta (2005). Equa­ tions (Q2), (Q3)$=i, (H2) and (H3)^= i appeared explicitly for the first time in Adler-Bobenko-Suris (2003). The master equation (Q4) was investigated in Adler-Suris (2004), where its relation to various 2D integrable systems was revealed. Special solutions to this equation were found in Atkinson-Hietarinta-Nijhoff (2007). Symme­ tries of quad-equations from our lists were studied in Papageorgiou-TongasVeselov (2006) and Rasin-Hydon (2007). A 3D consistent equation without the tetrahedron property was found in Hietarinta (2004). This equation was shown to be linearizable by RamaniJoshi-Grammaticos-Tamizhmani (2006). Its geometric interpretation is given in Adler (2006).

Section 6.13: Integrable discrete Laplace type equations. The rela­ tion of discrete (hyperbolic) systems on quad-graphs to Laplace type (ellip­ tic) equations was discovered in Bobenko-Suris (2002). Examples of Laplace type equations on graphs previously appeared in Adler (2001). The threeleg forms of integrable quad-equations were found in Adler-Bobenko-Suris (2003) (with a formula for (Q4) in the Weierstrass normalization). In AdlerSuris (2004) the three-leg form of (Q4) was used to derive elliptic Toda systems on graphs. Section 6.14: Yang-Baxter maps were introduced in Drinfeld (1992) under the name of set-theoretical solutions of the Yang-Baxter equation. In Veselov (2003) the term “Yang-Baxter maps” was proposed instead of “set-theoretical solutions” , and various notions of integrability were stud­ ied. In particular, commuting monodromy maps were constructed and zero curvature representations were discussed. A general construction of zero curvature representations (Theorem 6.36) was given subsequently in SurisVeselov (2003). A good survey on the topic is by Veselov (2007). The map of Example 1 first appeared in Adler (1993). Example 2 is treated in Goncharenko-Veselov (2004) along with more general Yang-Baxter maps on Grassmannians. Example 3 is investigated in Noumi-Yamada (1998) and in Etingof (2003).

6.18. Bibliographical notes

289

Section 6.15: Classification of Yang-Baxter maps. Quadrirational Yang-Baxter maps were introduced and classified in Adler-Bobenko-Suris (2004). On pencils of conics used in this classification one can read, for example, in Berger (1987). Section 6.16: Discrete integrable 3D systems. Various algebraic struc­ tures relevant for integrability of higher-dimensional discrete systems ap­ peared in the literature. The role played in 2D by the zero curvature repre­ sentation goes in 3D to the so-called local Yang-Baxter equation introduced in Maillet-Nijhoff (1989). Several 3D systems possessing this structure were found in Kashaev (1996). The functional tetrahedron equation was intro­ duced in Kashaev-Korepanov-Sergeev (1998) as one of the versions of the 4D consistency. Note that their notation is different from the one in formula (6.136): their indices 1 < z, j, k < 6 of numerate two-dimensional coor­ dinate planes. This paper contains also a list of solutions of this equation possessing local Yang-Baxter representations with a certain Ansatz for the participating tensors. The discrete Darboux system was derived in Bogdanov-Konopelchenko (1995). The fact that the star-triangle map satisfies the functional tetra­ hedron equation was observed in Kashaev (1996). In discrete differential geometry the star-triangle map appeared in Konopelchenko-Schief (2002a). The discrete B K P equation goes back to Miwa (1982). Its double cross­ ratio form is due to Nimmo-Schief (1997). Its 4D consistency was observed in Adler-Bobenko-Suris (2003). Theorem 6.47 is due to Tsarev-Wolf (2007). The first works on quantization of discrete differential geometry ap­ peared recently. Quantum versions of the discrete Darboux system and its reduction for circular nets were investigated in Sergeev (2007) and BazhanovMangazeev-Sergeev (2008). A quantization of circle patterns is proposed in Bazhanov-Mangazeev-Sergeev (2007).

Section 6.17: Exercises. Ex. 6.8: This result is due to Adler-Sokolov; see Adler-Bobenko-Suris (2007). Ex. 6.9, 6.10, 6.11: See Adler-Bobenko-Suris (2003). Ex. 6.12: See Hietarinta (2004). Ex. 6.13: See Adler-Bobenko-Suris (2007). Ex. 6.14: See Adler-Bobenko-Suris (2007) and Atkinson (2008). Ex. 6.16: Unpublished result by Adler. Ex. 6.18: See Adler-Suris (2004). Ex. 6.19: In this generality the result seems to be new.

290

6. Consistency as Integrability

Ex. 6.20, 6.21: See Hoffmann (2008), Schief (2007), and Pinkall-Springborn-Weifimann (2007). Ex. 6.25: See Adler-Bobenko-Suris (2004). The map F\ appeared also in a different context in Tongas-Tsoubelis-Xenitidis (2001). Ex. 6.27: See Kashaev-Korepanov-Sergeev (1998). Ex. 6.28: (2005).

A related material can be found in Konopelchenko-Schief

Ex. 6.29: From a work in progress with Adler.

Chapter 1

Discrete Complex Analysis. Linear Theory

7.1. Basic notions of discrete linear complex analysis Many constructions in discrete complex analysis are parallel to discrete dif­ ferential geometry in the space of real dimension 2. Recall that a harmonic function u relation d2u

: R 2 ~

C

—> R

is characterized by the

d2u

dx* + d ^ = A conjugate harmonic function v : Riemann equations

R 2

~ C —> R is defined by the Cauchy-

dv

du

dv

du

dy

dx ’

dx

dy

Equivalently, / = u + iv : R 2 ~ C —> C is holomorphic, i.e., satisfies the Cauchy-Riemann equation d £ ^ .d l dy

%d x

The real and the imaginary parts of a holomorphic function are harmonic, and any real-valued harmonic function can be considered as a real part of a holomorphic function. A standard classical way to discretize these notions is the following. A function u : 1? —» R is called discrete harmonic if it satisfies the discrete Laplace equation ( A u ) m ^n =

U m + i^ n + U rn—

+

^771,71+1 H"

1 — 4 U m ,n =

0.

291

292

7. Discrete Linear Complex Analysis

A natural domain of a conjugate discrete harmonic function v : (Z 2)* —►R is the dual lattice ; see Figure 7.1. The defining discrete Cauchy-Riemann

f---------------- 1f---------------- 1

V— 1 1 >

1 1 1 —

—

T i

1i

1 i i

11

i *

- - 6

i---------------- 1I-----------------t F i g u r e 7.1. Regular square lattice and its dual.

equations read: ^771+1/2,7-1+1/2 — ^771+1/2,71— 1/2

—

^ r a + l , 7 i — ^ r a ,n ?

^771 + 1/2,71+1/2 ~

=

_ (^771,71+1 ~

^772-1/2,71-1/2

^771,77 ) ?

with the natural indexing of the dual lattice; cf. Figure 7.2.

?

Vi

#

u1

V0 Q - -

Vi -

Vo =

Ul -

• Uo

V \ -V Q

Ui

O

6 Vo =

The corre-

vi

UQ

-(t t l -

Uq)

Figure 7.2. Discrete Cauchy-Riemann equations in terms of u yv.

sponding discrete holomorphic function / : Z 2 U (Z 2)* —> C is defined on the superposition of the original square lattice Z 2 and the dual (Z 2)*, by the formula

which comes to replace the smooth version f = u + iv. Remarkably, the dis­ crete Cauchy-Riemann equation for / is one and the same for both pictures:

/ tti,71+ 1 /2 see Figure 7.3.

f m , n —1 /2

^(/ tti+ 1 / 2,71

f m —l/2,n)i

293

7.1. Basics o f discrete complex analysis

h ~ f 2 = i(fa ~ h )

F i g u r e 7.3. Discrete Cauchy-Riemann equations in terms of /.

This discretization of the Laplace and the Cauchy-Riemann equations apparently preserves the majority of important structural features. Its gen­ eralization for arbitrary graphs goes as follows. Discrete harmonic functions can be defined for an arbitrary graph S with the set of vertices V (S ) and the set of edges E ( S).

Definition 7.1. (Discrete Laplacian and discrete harmonic func­ tions) F o r a given weight function v : E ( S) —* M+ on edges o f the discrete Laplacian is the operator acting on functions f : F (S ) (7. 1)

(A/)(aro) = ^

v

(xq,

C by

x ) { f ( x ) - f ( x 0)),

X~Xo where the summation is extended over the set of vertices x connected to

xq

by an edge. A function f : V (S ) —> C is called discrete harmonic (with respect to the weights v ) if A / = 0.

The positivity of weights v in this definition is important from the ana­ lytic point of view, since it guarantees, e.g., the maximum principle for the discrete Laplacian under suitable boundary conditions (so that discrete har­ monic functions come as minimizers of a convex functional). However, from the pure algebraic point of view, one might consider at times also arbitrary real (or even complex) weights. If S comes from a cellular decomposition of an oriented surface, let S* be its dual graph, and let the quad-graph D be its double; see Section 6.4. Extend the weight function to the edges of S* according to the rule (7.2)

294

7. Discrete Linear Complex Analysis

Definition 7.2. (Discrete Cauchy-Riemann equations and discrete holomorphic functions) A function f : V(T>) —> C is called discrete holo­ morphic (with respect to the weights v ) if fo r any positively oriented quadri­ lateral (x o ,y o ,x i,y i) G F ( D ) (see Figure 7A ), tn o\ (7.3)

f ( V l )-— ~ f(V ■ {x , 0,x i)^ = —— - —0 ) = iu f ( x i ) - f { x 0) ’

1 iv (y 0, y i ) '

These equations are called the discrete Cauchy-Riemann equations.

Vo F i g u r e 7.4. Positively oriented quadrilateral, with a labelling of di­ rected edges.

The relation between discrete harmonic and discrete holomorphic func­ tions is the same as in the smooth case. It is given by the following statement, which is a special case of Theorem 6.31.

Theorem 7.3. (Relation between discrete harmonic and discrete holomorphic functions) a) I f a function f : V (D ) —> C is discrete holom orphic, then its restric­ tions to V (5 ) and to K (S *) are discrete harmonic.

b) Conversely, any discrete harmonic function f : V (5 ) —* C admits a family of discrete holomorphic extensions to V (*D ), differing by an additive constant on V^S*)- Such an extension is uniquely determined by a value at one arbitrary vertex y G V (S * ).

7.2. M ou tard transform ation for discrete Cauchy-Riem ann equations Observe that discrete Cauchy-Riemann equations (7.3) formally are not dif­ ferent from the Moutard equations (2.51) for T-nets. One only has to fix an orientation of all quadrilateral faces (xo, yo, x\, V i) £ F (T>). We assume that it is inherited from the orientation of the underlying surface. One can now apply the Moutard transformation of Section 2.3.9 to dis­ crete holomorphic functions. To this aim, one has to choose an orientation of all elementary quadrilaterals in Figure 7.5. This can be done, for example,

7.2. Moutard transformation for discrete Cauchy-Riemann equations

295

as follows: for the quadrilaterals (x q , y$ , x^ , y ± ) G F (D + ), choose an orien­ tation to coincide with that of the corresponding (#o, Vo, V i) G F (D ). For a “vertical” quadrilateral over an edge (x, y) G E ( D ), assume that x G V^S), y G V^S*), and choose the positive orientation corresponding to the cyclic order (x, y, y+ , x + ) of its vertices. Observe that under this convention, two opposite “vertical” quadrilaterals are always oriented differently.

In the case of arbitrary quad-graphs, one has to generalize one more ingredient of the Moutard transformation, namely the data (M T ^ ).

Theorem 7.4. (Moutard transformation for discrete holomorphic functions) On an arbitrary bipartite quad-graph D, valid in itial conditions fo r a Moutard transformation o f the discrete Cauchy-Riemann equations consist of

( M C R f ) the value of f + at one point x(0)

G

V (V );

(M C R ^ ) the values o f weights on “vertical” quadrilaterals (x, y, y + , x + ) assigned to all edges (x ,y ) o f a Cauchy path in V . See Theorem 6.6 for necessary and sufficient conditions for a path to be a Cauchy path, i.e., to support initial data for a well-posed Cauchy problem. It is natural to assign the weights on the “vertical” quadrilaterals to the underlying edges of D. Weights v on the faces of D together with the data (M C R ^) yield the transformed weights v + on the faces of D + , as well as the weights over all edges of E ( D). This can be considered as a Moutard transformation for the Cauchy-Riemann equations on D. Finding a solution f : V (T )+ ) —> C of the transformed equations requires additionally the datum (M C R f). Note that the system of weights v is highly redundant, due to (7.2). To fix the ideas in writing the equations, we stick to the weights assigned to

296

7. Discrete Linear Complex Analysis

the “black” diagonals of the quadrilateral faces of the complex D. On the ground floor, these are the edges of the “black” graph S; on the first floor, these are the edges of the “black” graph which is a copy of S*; and for the “vertical” faces, these are the edges (x ,y + ), with x G V (S ) and y G V^S*). Needless to say that the latter weights can be assigned to the quad-graph edges (x ,y ) G E ( D). So, we write the discrete Cauchy-Riemann equations as followrs: (7.4)

f(y i)~ f(y o )

(7.5)

f(xt)-f(x t)

(7.6)

f(x+)~ f{y )

=

i v ( x o , x i ) ( f ( x i ) - / (x 0)),

= ir'(y£,vt)(nvt)-f(yo))> =

iv(x,y+ ) ( f ( y + ) - f( x )).

Denote, for the sake of brevity, v = i/{x0, x i ) ,

V+ = v { y £ , y t ),

= u (x j, y £ ).

Regarding the weights v , /ioo, and ^oi as the input of the Moutard transfor­ mation on an elementary hexahedron of D, its output consists of the weights z/+, /.iio, and /xn, given by (cf. (2.59)) (7.7)

v ^ v = -/in/ioo = -MioMoi = -----------------• Moo ~ Moi - v

This transformation is well defined for real weights v , fij^, but it does not preserve, in general, positivity of the weights v . To give a different form of this transformation, observe that the relation MuMoo — MioMoi for each elementary quadrilateral (xo, yo, xi, y i) of D yields the existence of the function 9 : V (D ) —> C, defined up to a constant factor, such that — 6 (y k )/ 0 (x j) (see Exercise 7.1). Moreover, choosing 0(xo) real at some point xo G V'(S), one sees that 9 takes real values on V (S ) and imaginary values on V (S * )- An easy computation shows that the last equation in (7.7) is equivalent to 0 (yi) - % o ) = iv (x Q ,x i)(d { x i) - 0(xo)), so that the function 9 is discrete holomorphic with respect to the weights v. For the transformed weights one finds:

+ _ Q(yo)0 (vi) (78>

" 1,- « ( x „ ) « ( x 1) '

Conversely, an arbitrary discrete holomorphic function 9 : V(T>) —> C de­ fines, via (7.8), a Moutard transformation of the discrete Cauchy-Riemann equations. It should be mentioned that the data (M C R ^ ) can be reformu­ lated in terms of the function 9: (M C R ^ )

the values of 9 at all vertices along a Cauchy path in D.

297

7.3. Integrable discrete Cauchy-Riemann equations

Remark. A Moutard transformation for discrete Cauchy-Riemann equa­ tions yields, by restriction to the “black” graphs, a sort of Darboux trans­ formation of arbitrary discrete Laplacians on S into discrete Laplacians on

s*. 7.3. Integrable discrete Cauchy-Riem ann equations We now turn to a useful question of “stationary points” of the Moutard transformation discussed in the previous section. More precisely, this is the question about conditions on the weights v : E ($ ) —> R+ such that there exists a Moutard transformation for which the opposite faces of any elementary hexahedron of D (see Figure 7.5) carry identical equations.

Theorem 7.5. (Integrability of discrete Cauchy-Riemann equations) A system o f discrete Cauchy-Riemann equations with the weight function v : E ( S) U i?(9*) —►R+ satisfying (7.2) admits a Moutard transformation into itself if and only if fo r all x q G ^ (S ) and all yo G F (S *) the following conditions are fulfilled:

TT

(7 9 ) 1 j

11

1 — iu {e )

=

eE star(xo;3)

i

’

TT

-± M

^1 = 1 l - i v ( e*)

AI

e*£ sta r(y o ;S *)

Proof. Opposite faces of D and

carry identical equations if v + v = 1 in (7.7). Clearly, this yields also /in/ioo = MioMoi — —1? which means that the opposite “vertical” faces also support identical equations (recall that opposite “vertical” faces carry different orientations). Moreover, given v — v (x q ,x \ ) for an elementary quadrilateral (xQ ,yQ ,x\,yi) of 2), we find that the input data /ioo, Hoi of the Moutard transformation should be related as follows:

Moo

—

Ml

i = 1 “

V

^ <=>

Moo ~ v fl fioi — ------- — = ( , /j L q q U

+1

-v \ r 1 -i I [ m b

1

where the standard notation for the action of P G L (2 ,C ) on C by Mobius transformations is used. This means that all the weights on the vertical faces of a “stationary” Moutard transformation are completely defined by just one of them, so that such transformations form a one-parameter family. To derive a condition for v for the existence of a “stationary” Moutard transformation, consider a flower of quadrilaterals (#o, V k-h Vk) around G F (S ) (see Figure 6.5). In the natural notation, we find: x q

Mo,fc-i “ vk (1 M o , = ------------—r = I WU-i^fc + l \"k

~Vk\ r

i , lW),fc—l j • 1 J ’

298

7. Discrete Linear Complex Analysis

Running around xo should for any /ioo return its value, which means that the matrix product

should be proportional to the identity matrix. This matrix product is easily computed (see Exercise 7.2): a

=

k

b

k

= ^ ( n a + ^ - n a - * * ) ) . k

k

and the condition B — 0 is equivalent to the first equality in (7.9). The sec­ ond condition in (7.9) is proved similarly, by considering a flower of quadri­ laterals around yo G V(S*)d Thus, the existence of a “stationary” Moutard transformation singles out a special class of discrete Cauchy-Riemann equations, which have to be con­ sidered as 2D systems with the 3D consistency property; see Section 6.7. In other words, such Cauchy-Riemann equations should be termed integrable. The main difference as compared with the examples in Section 6.7 is that discrete Cauchy-Riemann equations naturally depend on the orientation of the elementary quadrilaterals, and that their parameters v are apparently assigned not to the edges of the quad-graph, but rather to the diagonals of its faces. The integrability condition (7.9) admits a nice geometric interpretation. It is convenient (especially for positive real-valued v ) to use the notation (7.10)

i/ (e )= t a n ^ ^ , z

0(e)G(O,7r).

The condition v (e *) = 1/ v(e) is translated into (7.11)

(e*) = 7T - 0(e),

while the condition (7.9) says that for all xo G V'(S) and all yo G V (9 *), (7.12)

Y[

exp(z0(e)) = 1,

e€ star(zo;S )

JJ

exp (i(e*)) = 1.

e*€ sta r(y o ;S *)

These conditions should be compared with conditions characterizing the angles 0 : E ( 9) U £7(9*) —►(0 ,7r) of a rhombic embedding of a quad-graph 2), which consist of (7.11) and (7-13)

^ eE star(xo;S)

4>(e) = 2tr,

^

= 2tt,

e*G star(yo;SJ ")

for all xq 6 V (S ) and all yo € Vr(S*)- Thus, the integrability condition (7.12) says that the system of angles (p : E (S ) U E ( S*) —►(0,7r) comes from

299

7.3. Integrable discrete Cauchy-Riemann equations

a realization of the quad-graph D as a rhombic ramified embedding in C. Flowers of such an embedding can wind around its vertices more than once. Another formulation of the integrability conditions is given in terms of the edges of the rhombic realizations.

Theorem 7.6. (Integrable Cauchy-Riemann equations in terms of rhombic edges) Integrability condition (7.9) fo r the weight function v : E (S ) U E (S * ) —> M+ is equivalent to the following: there exists a labelling of directed edges o f V , 9 : E(T>) —> S1, such that, in the notation o f Figure 7A,

(7-14)

v (x 0, x i ) = — - ^ --- r = t

v{yo,yi)

0o + 0i

.

Under this condition, the 3D consistency o f the discrete Cauchy-Riemann equations is assured by the following values o f the weights v on the diagonals o f the vertical faces o f D :

(7.15)

+'i) v { x ,y +

-

9- A

0 + A’

where 9 = 0 (x ,y ), and A £ C is an arbitrary number which is interpreted as the label assigned to all vertical edges o f D : A = 9 (x ,x + ) = 9 (y ,y + ).

So, integrable discrete Cauchy-Riemann equations can be written in a form with parameters assigned to directed edges of 2): /7 1 f i x ;

f ( y i ) - f ( y o ) .. 0i - 0 q f ( x i ) - f ( x 0)

0i + 0o ’

where 0o = p(yo) ~ p(xo) = p(xi) - p( yi) ,

0i = p{ yi) - p( x0) = p(xi) - p( y0),

and p : V (S ) —■ ►C is a rhombic realization of the quad-graph D. Since 0i -

Oo

di + 90

=

p(yi) -

pjyo

)

p (x i) - p ( x 0) ’

we see that for a discrete holomorphic function f : V ( S) C, the quotient of diagonals of the /-image of any quadrilateral (a?o, 2/0, # i 5y i) ^ F ( V ) is equal to the quotient of diagonals of the corresponding rhombus. A standard construction of zero curvature representation for 3D con­ sistent equations, given in Theorem 6.4, leads in the present case to the following result.

Theorem 7.7. (Zero curvature representation of discrete CauchyRiemann equations) The discrete Cauchy-Riemann equations (7.16) ad­ m it a zero curvature representation with spectral parameter dependent 2 x 2

7. Discrete Linear Complex Analysis

300

matrices along (x, y ) G jE7(D ) g w e n by , - 17.

(7.17)

/a + 0

L(y, a:, a; A) =

-2 0 (/ (* ) + f(y))\

x-e

y o

J

,

where 9 = p (y ) —p (x ).

Linearity of the discrete Cauchy-Riemann equations is reflected in the triangular structure of the transition matrices. Also, all constructions of Section 6.8 can be applied to integrable dis­ crete Cauchy-Riemann equations. In particular, for weights coming from a quasicrystallic rhombic embedding of the quad-graph D, with labels © = { ± 01 , . . . , ± Qd}, discrete holomorphic functions can be extended from the corresponding surface C Z d to its hull, preserving discrete holomorphy. Here we have in mind the following natural definition:

Definition 7.8. (Discrete holomorphic functions on Z d) A function f : Z d —> C is called discrete holomorphic if it satisfies, on each elementary square o f Z d, the equation

/(re + ej + ek) - f ( n ) = 0j + 6k

(7 18v

'

f ( n + ej) - f ( n + ek)

9j-0k'

For discrete holomorphic functions in Z d, the transition matrices along the edges (re, re + ek) of Z d are given by (7.19)

L „ ( » ; A ) = ( A + I,‘!

\

- 2M / ( » + e *) + / ( » ) ) V

A — 0*

o

J

All results of this section hold also in the case of generic complex weights i/, which leads to 0 £ C and to parallelogram realizations of D.

7.4. Discrete exponential functions An important class of discrete holomorphic functions is built by discrete exponential functions. We define them for an arbitrary rhombic embedding p : V { V ) —> C. Fix a point xo G V^ID). For any other point x G V ( T > ) , choose some path { and x n Then

=

x.

E ( D ) connecting zo to x, so that tj = ( x j - i , x j )

Let the slope of the

(

\

j-

th edge be 9 j

= p(xj) — p (x j-i )

G S 1.

TI z j =l

J

Clearly, this definition depends on the choice of the point xo not on the path connecting xq to x.

G

V (T )), but

301

7.4. Discrete exponential functions

An extension of the discrete exponential function from f 1<e> to the whole of Iud is given by the following simple formula:

<™>

«<•**>- n k= 1

*

The discrete Cauchy-Riemann equations for the discrete exponential func­ tion are easily checked: they are equivalent to a sir^ple identity i

/£ + 0j

z + Ok _ A / f z + Qj _ z +

\z — Oj

z — Ok

) / \z — Oj

\ _ Oj + Ok

z — djk/

Oj — Ok

At a given n £ Z d, the discrete exponential function is rational with respect to the parameter z, with poles at the poiptti £\0\,. .. , where 6k = sign rife. Equivalently, one can identify the discrete exponential function by its initial values on the axes: (7.21)

e (n e k-z ) =

fz + Ok'

h'

Another characterization says that e(-;z) is the Backlund transformation of the zero solution of discrete Cauchy-Riemann equations on Z d, with the “vertical” parameter z. We now show that the discrete exponential functions form a basis in some natural class of functions (growing not faster than exponentially).

Theorem 7.9. (Discrete exponentials form a basis of discrete holo­ morphic functions) Let f be a discrete holomorphic function on V(T>) ~ V(£1'd), satisfying

(7.22)

|/(n)| < exp(C(|ni| H---- + M ) ) ,

with some

C

g

K.

Vn

G

V^Ad),

Extend it to a discrete holomorphic function on the hull

3-C(F(f]£>)). There exists a function g defined on the disjoint union o f small neighborhoods around the points ±0k G C and holomorphic on each o f these neighborhoods, such that

(7.23)

f ( n ) - /(0 ) = ^

. f 5 (A)e(n; X)dX,

Vn € 5C(V(J2®)),

where Y is a collection o f 2d small loops, each running counterclockwise around one o f the points ± 0 k .

Proof. The proof is constructive and consists of three steps. (i) Extend / from V (flx>) to ^ (V ^ fix ))); inequality (7.22) propagates in the extension process, if the constant C is chosen large enough.

302

7. Discrete Linear Complex Analysis

(ii) Introduce the restrictions dinate axes: fnk) = f ( n e k),

of / : 3 {(F (fii> )) —> C to the coor­

ak(ftT>) < n <

bk{ n v ) .

(iii) Set g ( A) = Y lk= i(9 k W + 9 - k W ) , where the functions g ± k( A) van­ ish everywhere except in small neighborhoods of the points d=0 respectively, and are given there by convergent series (7.24)

9 t i x) = ±

( / « - m

+ £

- e \ )),

and a similar formula for g_/-(A). Formula (7.23) is then easily verified by computing the residues at A = ± 9 k (see Exercise 7.5).

□ (k)

It is important to observe that the data fn , necessary for the con­ struction of g(A), are not among the values of / on V(X>) ~ V(Qx>) known initially, but are encoded in the extension process.

7.5. Discrete logarithmic function We now define the discrete logarithmic function on a rhombic quad-graph D. Fix some point xo G V^D), and set (7.25)

i ( x ) ^ ± . J ^ ^ l e( x ; x) d\,

V xgF(D ).

Here the integration path T is the same as in Theorem 7.9, and fixing xo is necessary for the definition of the discrete exponential function on D. To make (7.25) a valid definition, one must specify a branch of log(A) in a neighborhood of each point ±0^- This choice depends on x, and is done as follows. Assume, without loss of generality, that the circular order of the points ±9k on the positively oriented unit circle S 1 is the following: —01 , . . . , —0^. We set 0fc+d = ~0fc for fc = 1,.. ., d, and then define 9r for all r G Z by 2d-periodicity. For each r G Z, assign to 9r = exp(i 7 r) G S 1 a certain value of the argument G M: choose a value 71 of the argument of 01 arbitrarily, and then extend it according to the rule 7 r+l

- 7 r G (0 , 7r) ,

Vr

G

Z.

Clearly, 7 r+d — 7 r + 7r, and therefore also j r+2d — 7r + 27r. It will be convenient to consider the points 0r , supplied with the arguments 7 r , as belonging to the Riemann surface A of the logarithmic function (a branched covering of the complex A-plane).

303

7.5. Discrete logarithmic function

For each rn G Z, define the “sector” Um on the plane C carrying the quad-graph D as the set of all points of V(*D) which can be reached from xq along paths with all edges from { 0m, . . . , 0m+d-\ }• Two sectors Umi and Um2 have a nonempty intersection if and only if \m\ — m2|< d. The union U ~ Um=-oo Um is a branched covering of the quad-graph D, and it serves as the domain of the discrete logarithmic function. The definition (7.25) of the latter should be read as follows: for x G t/m, the poles of e(x; X) are exactly the points 0m, . . ., 0m+d-i £ A. The integration path T consists of d small loops on A around these points, and arg(A) = S lo g(A ) takes values in a small open neighborhood (in R) of the interval (7.26)

[7m, lm +d-l\

of length less than n. If m increases by 2d, the interval (7.26) is shifted by 27r. As a consequence, the function £ is discrete holomorphic, and its restriction to the set V^S) of “black” points is discrete harmonic everywhere on U except at the point .xo: (7.27)

A £ (x ) = 6X0X.

Thus, the functions gk in the integral representation (7.23) of an arbi­ trary discrete holomorphic function, defined originally in disjoint neighbor­ hoods of the points a r, in the case of the discrete logarithmic function are actually restrictions of a single analytic function log(A)/(2A) to these neigh­ borhoods. This allows one to deform the integration path T into a connected contour lying on a single leaf of the Riemann surface of the logarithm, and then use standard methods of complex analysis to obtain asymptotic ex­ pressions for the discrete logarithmic function. In particular, one can show that at the “black” points of F (S ), (7.28)

£ (x) ~ log \x — £()|,

x —> oo.

Properties (7.27), (7.28) characterize the discrete Green's function on S. Thus: T h e o re m 7.10. (D is cre te G re e n ’s fu n ction ) The discrete logarithmic function on D, restricted to the set o f vertices Vr(S) of the “black” graph S, coincides with discrete Green7s function on S.

Now we extend the discrete logarithmic function to Z rf, which will allow us to gain significant additional information about it. In addition to the unit vectors G Z d (corresponding to 0^ G S1), we introduce their opposites efc_|-d = —e/e, k G [1, d] (corresponding to O ^d = ~@k)i and define e r for all

304

7. Discrete Linear Complex Analysis

r G Z by 2d-periodicity. Then m + d —1

(7.29)

Sm =

0

Zer c Z d

r=m

is a d-dimensional octant containing exactly the part of f2x> which is the P-image of the sector Um C D. Clearly, only 2d different octants appear among the Sm (out of 2d possible d-dimensional octants). Define 5m as the octant Sm equipped with the interval (7.26) of values for S lo g(# r). By definition, Smi and Sm2 intersect if the underlying octants Smi and Sm2 have a nonempty intersection spanned by the common coordinate semiaxes Z er , and the Qlog(0r) for these common semiaxes match. It is easy to see that Smi and Sm2 intersect if and only if \m\ — m 2 \ < d . The union

$ = Um=—00Srn

a branched covering of the set

|Jm=i

D efin ition 7.11. (D is cre te logarith m ic function on

C

Zd.

Zd)

The discrete

logarithmic function on S is given by the form ula

(7.30)

l0^ A^ e ( n ; X)d\,

£ (n ) =

Vn G S,

where fo r n G Sm the integration path T consists o f d loops around the points 0m, . .. , 1 on A, and Q log(A) on T is chosen in a small open neighborhood o f the interval (7.26).

The discrete logarithmic function on D can be described as the restric­ tion of the discrete logarithmic function on S' to a branched covering of ~ D. This holds for an arbitrary quasicrystallic quad-graph D whose set of edge slopes coincides with 0 = { ± 0 i , . . ., ±0^}. Now we are in a position to give an alternative definition of the discrete logarithmic function. Clearly, it is completely characterized by its values £ (ner ), r G [m, m + d — 1], on the coordinate semiaxes of an arbitrary octant Sm. Let us stress once more that the points n e r do not lie, in general, on the original quad-surface

T h e o re m 7.12. (V alues o f discrete logarith m ic function on c o o rd i­ nate axes) The values £ $ — £(ner ), r G [m,m + d — 1], o f the discrete logarithmic function on Sm

(7.31)

f(v)

—

C

S are given by:

2 ( ! + 3 + •■• + + log(0r) = iy r ,

) ’

« even, n odd.

Here the values log(0r) = i j r are chosen in the interval (7.26).

P ro o f. Comparing formula (7.30) with (7.24), we see that the values £n ^ can be obtained from the expansion of log(A) in a neighborhood of A = 9r

7.5. Discrete logarithmic function

305

into the power series with respect to the powers of (A — 0r )/(X + 0r ). This expansion reads:

n= 1

Thus, we come to a simple difference equation

(7.32)

=

with the initial conditions (7.33)

t £r) = £(0) = 0,

4 r) = £(er ) = log(0r),

which yield (7.31).

□

Observe that values (7.31) at even (resp. odd) points imitate the be­ havior of the real (resp. imaginary) part of the function log(A) along the half-lines arg(A) = arg(0r). This can be easily extended to the whole of S. Restricted to the black points n G S (those with n\ + • • • +

even), the discrete logarithmic function models the real part of the logarithm. In particular, it is real-valued and does not branch: its values on Sm depend on m (mod 2d) only. In other words, it is a well-defined function on Sm. On the contrary, the discrete logarithmic function restricted to the white points n G S (those with n\ + • • • + rid odd) takes purely imaginary values, and increases by 2iri as m increases by 2d. Hence, this restricted function models the imaginary part of the logarithm. It turns out that recurrence relations (7.32) are characteristic for an im­ portant class of solutions of the discrete Cauchy-Riemann equations, namely for the isomonodromic solutions. In order to introduce this class, recall that discrete holomorphic functions in Z d possess a zero curvature representa­ tion with transition matrices (7.19). The moving frame ^(-,A) : Z d —» G L(2,C)[A] is defined by prescribing some \I/(0; A), and by extending it re­ currently according to the formula (7.34)

^ ( n + e k; A) = L k(n ; A)\t(n; A).

Finally, define the matrices A (-; A) : Z d —> gl(2,C)[A] by (7.35)

A(n-,X) =

a\

A-} ^ 1(n; A).

These matrices satisfy a recurrence relation, which is obtained by differen­ tiating (7.34), (7.36)

A ( n + ek; A) = dLk^

^ L ^ i n - A) + L k(n; A)A(n\ A )L ^ (n ; A),

and therefore they are determined uniquely upon fixing some A(0; A).

7. Discrete Linear Complex Analysis

306

Definition 7.13. (Isomonodromy) A discrete holom orphic function f : Z d —>C is called isomonodromic i f fo r some choice o f A ( 0; A), the matrices A ( n ; A) are meromorphic in A; with poles whose positions and orders do not depend on n E Z d.

This term originates in the theory of integrable nonlinear differential equations, where it is used for solutions with a similar analytic characteri­ zation. It is clear how to extend Definition 7.13 to functions on the covering S. In the following statement, we restrict ourselves to the octant S\ = ( Z + ) d for notational simplicity.

Theorem 7.14. (Discrete logarithmic function is isomonodromic) F or a proper choice of A ( 0; A), the matrices A ( n ; A) at any point n E (Z + )d have simple poles only: ,

v

(7.37)

^

A ^ (n )

A( „ ; * ) = _ A J

(B ^ l\ n )

+ £

( ^

i

C ^ (n )\

+ ^ 2 ) .

L

1 = 1

1

with

.0 (7.38)

^ ° )(n )

(_1Y n i+ -+ n d 1 j 0

= ,0

m B^(n)

(7.39)

=

n,

i 1 - ( ( ( n ) + £(n - et)) V

.0

(7.40)

C ‘‘>(n)

=

n, V

U!

0

°

'< " + e ‘ > + '< ">

°

1

A t any point n E S\ the following constraint holds: d

(7.41)

^ 1= 1

m (^({n

+

ei)

—t { n

—

e ;)) = 1 - ( - 1 ) ” 1+

" +nd

Proof. The proper choice of A(0; A) mentioned in the Theorem, can be read off formula (7.38):

4(0; A)

1 A)

A

r

0

The proof consists of two parts. (i) First, one proves the claim for the points of the coordinate semi­ axes. For any r — 1,..., d, construct the matrices A (n e r ; A) along

307

7.6. Exercises

the r-th coordinate semi-axis via formula (7.36) with transition ma­ trices (7.19). This formula shows that the singularities of A (n e r \A) are poles at A = 0 and at A = =L0r , and that the pole A = 0 re­ mains simple for all n > 0. By a direct computation and induc­ tion, one shows that it is exactly the recurrence relation (7.32) for fn^ — f ( n e r ) which assures that the poles A = ± 0 r remain simple for all n > 0. Thus, (7.37) holds on the r-th coordinate semiaxis,

with B^l\ n e r ) = C ® (n e r ) = 0 for I / r. (ii) The second part of the proof is conceptual, and is based upon the multidimensional consistency only. Proceed by induction, filling out the hull of the coordinate semiaxes: each new point is of the form n + ej + e*, j ^ fc, with three points n, n + e j, and n + known from the previous steps, where the statements of the proposition are assumed to hold. Suppose that (7.37) holds at n + ej, n + e*;. The new matrix A ( n + ej + e^; A) is obtained by two alternative formulas, (7.42)

A ( n + ej + ek\A) =

L kl ( n + ev

+ L k( n + ej\ A) A ( n + ey, A) L ^ 1( n + ey, A), and the other with k and j interchanged. Equation (7.42) shows that all poles of A ( n + ej + e^; A) remain simple, with the possible exception of A = ±0^, whose orders might increase by 1. The same statement holds with k replaced by j . Therefore, all poles remain simple, and (7.37) holds at n + ej + ek. Formulas (7.38)-(7.40) and constraint (7.41) follow by direct computations based on (7.42). □

7.6. Exercises 7.1. Let 2) be a bipartite quad-graph, with black vertices x j and white vertices yj. Let ji : E(T>) —►C be a function such that, for any elementary quadrilateral (#o, yo>

U i) € F ( (D ) 1

li(xo,yQ) n{ x i , y i ) = fJ.(xo,yi)n(xi,y0). Show that there exists a function 0 : V (D ) —> C such that for every edge ( x, y) G E(*D ) we have i/jL(x,y) = 6 (y )/ 9 (x ). If fi is real-valued, then one can assume that 0 takes real values at black points and imaginary values at white points. 7.2. Prove by induction that the entries of the matrix

308

7. Discrete Linear Complex Analysis

are given by

k

k

k

k

7.3. Check that the function / : 1? —> C given by /(ra, n ) = (ra0i + n02)2 satisfies the discrete Cauchy-Riemann equation f ( m + l , n + 1) - /(m ,n ) ^ 0i + 02

/(m + 1, n) - /(m, n + 1)

0X - 02'

Generalize this function ( “discrete z2v) for Z d and for arbitrary quad-graphs 2).

7.4. Find the “discrete z 3” , i.e., the function / : Z 2 —> C which is polynomial in ra, n of degree 3, with cubic terms ( m6\ +n02)3, and satisfying the discrete Cauchy-Riemann equations.

7.5. Prove that for the functions gk{A) from (7.24), ReSA= ^

7.6. Estimate the difference

9k^

=

c) ~

'

— lo g n for the values given in (7.31), for n

even.

7.7. Bibliographical notes Section 7.1: Basic notions of discrete linear complex analysis. The standard discretization of harmonic and holomorphic functions on the reg­ ular square grid goes back to Ferrand (1944) and Duffin (1956). This dis­ cretization of the Cauchy-Riemann equations apparently preserves the ma­ jority of important structural features. A pioneering step in the direction of further generalization of the notions of discrete harmonic and discrete holomorphic functions was undertaken by Duffin (1968), where the combi­ natorics of Z 2 was given up in favor of arbitrary planar graphs with rhombic faces. A far reaching generalization of these ideas was given by Mercat (2001), who extended the theory to discrete Riemann surfaces. Section 7.2: Moutard transformation for discrete Cauchy-Riemann equations. For general Moutard transformations see the bibliographical note to Section 2.3 and Exercise 2.27. A further discussion of the Darboux transformation for discrete Laplace operators induced by the Moutard trans­ formation for discrete Cauchy-Riemann equations can be found in DoliwaGrinevich-Nieszporski-Santini (2007).

Section 7.3: Integrable discrete Cauchy-Riemann equations. Con­ dition (7.13) on the system of angles (j) : E ( 9) U £7(3*) —* (0,7r) character­ izing rhombic embedding was given in Kenyon-Schlenker (2004). Theorems

7.7. Bibliographical notes

309

7.5, 7.6 characterizing 3D consistent (integrable) Cauchy-Riemann equa­ tions and their zero curvature representation from Theorem 7.7 are from Bobenko-Mercat-Suris (2005).

Section 7.4: Discrete exponential functions. A discrete exponential function on Z 2 was defined and studied in Ferrand (1944) and DufEn (1956). It was generalized for quad-graphs D in Mercat (2001) and Kenyon (2002). The question whether discrete exponential functions form a basis in the space of discrete holomorphic functions on 2) (Theorem 7.9) was posed in Kenyon (2002) and answered in Bobenko-Mercat-Suris (2005).

Section 7.5: Discrete logarithmic function. The discrete logarith­ mic function on a rhombic quad-graph 2) was introduced in Kenyon (2002). Also the asymptotics (7.28) as well as Theorem 7.10 were proven in that paper. All other results in this section, starting with the extension of the discrete logarithmic function to Z d, are from Bobenko-Mercat-Suris (2005). For the theory of isomonodromic solutions of differential equations and its application to integrable systems see Fokas-Its-Kapaev-Novokshenov (2006). Isomonodromic constraint (7.41) was found in Nijhoff-RamaniGrammaticos-Ohta (2001), with no relation to the discrete logarithmic func­ tion.

Chapter 8

Discrete Complex Analysis. Integrable Circle Patterns

8.1. Circle patterns The idea that circle packings and, more generally, circle patterns serve as a discrete counterpart of analytic functions is by now well established. We give here a presentation of several results in this area, which treat the inter­ relations between circle patterns and integrable systems.

D efin itio n 8.1. (C irc le p a tte rn ) Let S be an arbitrary cell decomposition o f an open or closed disk in C. A map z : V'(S) —►C defines a circle pattern with combinatorics of S if the following condition is satisfied. Let y G F ( S ) — V(S* ) be an arbitrary face o f S, and let x \ ,x 2, . . . , x n be its consecutive vertices. Then the points z (x i ), z (x 2) , . • •, z (x n) G C lie on a circle , and their circular order is just the listed one. We denote this circle by C (y ), thus putting it into a correspondence with the face y, or, equivalently, with the respective vertex o f the dual cell decomposition S*.

As a consequence of this condition, if two faces yo,yi G F ( S) have a common edge (x q ,x \ ), then the circles C (y o) and C (y\) intersect in the points z (x i), z (x 2). In other words, the edges from E ( S) correspond to pairs of neighboring (intersecting) circles of the pattern. Similarly, if several faces 2/1>2/2? • • • ,Vm G F ( S) meet in one point xo G V^S), then the corresponding circles C (y \ ), C (y 2), •.., C (y m) also have a common intersection point z (x 0). A finite piece of a circle pattern is shown in Figure 8.1.

311

312

8. Integrable Circle Patterns

Figure 8.1. Circle pattern.

Given a circle pattern with combinatorics of S, we can extend the func­ tion z to the vertices of the dual graph, setting z (y ) = center of the circle C (y),

y € F ( S) — V^S*)-

After this extension, the map z is defined on all of V ( V ) — V (S) U V (S * ), where D is the double of 9- Consider a face of the double. Its 2-image is a quadrilateral of the kite form, whose vertices correspond to the intersection points and the centers of two neighboring circles Co, C\ of the pattern. De­ note the radii of Co,Ci by r*o,ri, respectively. Let x$,x\ correspond to the intersection points, and let yo^yi correspond to the centers of the circles. Give the circles Co, C\ a positive orientation (induced by the orientation of the underlying C), and let (j) G (0 , i r) stand for the intersection angle of these oriented circles. This angle is equal to the kite angles at the “black” ver­ tices z (x o), z (x i); see Figure 8.2, where the complementary angle 4>* — i r — is also shown. It will be convenient to assign the intersection angle = (e) to the “black” edge e = (x o ,x \ ) G E ( S), and to assign the complementary angle 0* = (e*) to the dual “white” edge e* = (y o ,y i) G E (S * )- Thus, the function cj) : E ( S) U E ( S*) —> (0,7r) satisfies (7.11). The geometry of Figure 8.2 yields following relations. First of all, the cross-ratio of the four points corresponding to the vertices of a quadrilateral face of D is expressed through the intersection angle of the circles Co, C\\ (8.1)

q (z (x 0), z(y 0) , z { x i ) , z ( y i ) ) = exp(2**).

8.2. Integrable cross-ratio and Hirota systems

313

Furthermore, running around a “black” vertex of D (a common intersection point of several circles of the pattern), we see that the sum of the consecutive kite angles vanishes (mod 27r), hence: (8.2)

exp(i(e)) = 1,

Va;0 € F (S ).

eGstar(xo;S)

Finally, let ^oi be the angle of the kite (z (x $ ), z(yo), z(x\ ), z(y \ )) at the “white” vertex z(y o ), i.e., the angle between the half-lines from the center z(yo) of the circle Co to the intersection points z ( xq ), z(x\ ) with its circle C\. It is not difficult to calculate this angle: /CQ>, (8-3)

r i \ ro + n e x p ^ * ) exp(i^oi) = ----:--------- , ro + n exp

Running around the “white” vertex of D, we come to the relation -A- r0 + r 7-exp(i*)

(8'4>

nro+ r ; e x p ( - 4 ) = 1 -

VW£1,(S' )'

where the product is extended over all edges e* = (y o ,y j) € star(yo; S*)> and * = (e*), while Tj are the radii of the circles C j — C (y j). 8 .2 . Integrable cross-ratio and H irota systems Our main interest is in the circle patterns with prescribed combinatorics and with prescribed intersection angles for all pairs of neighboring angles. According to formula (8.1), prescribing all intersection angles amounts to prescribing cross-ratios for all quadrilateral faces of the quad-graph V . Thus, we come to the study of cross-ratio equations on arbitrary quad-graphs.

314

8. Integrable Circle Patterns

Let there be given a function Q : E (S ) U E ( S*) —►C satisfying the condition (8.5)

Q (e *) = 1/Q(e),

Ve € E (S ).

Definition 8.2. (Cross-ratio system) The cross-ratio system on D corre­ sponding to the function Q consists of the following equations fo r a function z : V(T>) —> C, one fo r any quadrilateral face (xo, yo, £i?2/i) o f V :

(8.6)

q (z {x 0), z (y 0) , z ( x l ),z {y -i)) = Q {x 0, x i ) = l/Q(yo,yi).

An important distinction from the discrete Cauchy-Riemann equations is that the cross-ratio equations actually do not depend on the orientation of quadrilaterals. We have already encountered 3D consistent cross-ratio systems on Z d in Section 6.7 (see equation (6.33)), in the version with labelled edges. A natural generalization to the case of arbitrary quad-graphs is this: Vi

Vo

F i g u r e 8.3. Quadrilateral, with a labelling of undirected edges.

Definition 8.3. (Integrable cross-ratio system) A cross-ratio system is called integrable if there exists a labelling a : E(*D ) —> C o f undirected edges of V such that the function Q admits the following factorization (in the notation o f Figure 8.3):

(8.7)

Q ( x 0, x i ) = — -----r = — •

<3(2/0, yi)

ai

Clearly, integrable cross-ratio systems are 3D consistent (see Theorem 4.26), admit Backlund transformations , and possess zero curvature repre­ sentation with the transition matrices (6.47). It is not difficult to give an equivalent reformulation of the integrability condition (8.7).

Theorem 8.4. (Integrability condition of a cross-ratio system) A cross-ratio system with the function Q : E ( S) U E ( S*) —> C is integrable if

315

8.2. Integrable cross-ratio and Hirota systems

and only i f fo r all xq

G

V (S ) and fo r all yo

G

V (S * ) the following conditions

are fulfilled:

(8.8)

J]

Q (e ) = 1,

eE star(xo;S)

n

Q (e * ) = 1.

e*€star(t/o;S*)

For a labelling of undirected edges a : E ( V ) —> C, we can find a la­ belling 6 : E(*D) —►C of directed edges such that a = 92. The function p : V ( V ) —> C defined by p (y ) — p (x ) = 0 (x ,y ) gives, according to (8.8), a

parallelogram realization (ramified embedding) of the quad-graph D. The cross-ratio equations are written as q2

(8.9)

q ( z ( x o), z(yo), z ( x i), z{y1)) =

= q ( p ( x 0) , p ( y o ) , p ( x i ) , p ( y i ) ) ;

in other words, for any quadrilateral (#o,yo, V i) ^ the cross-ratio of the vertices of its image under the map z is equal to the cross-ratio of the vertices of the corresponding parallelogram. In particular, one always has the trivial solution z (x ) = p (x ) for all x G V(T>). A very useful transformation of the cross-ratio system is given by the following construction.

Definition 8.5. (Hirota system) F or a given labelling o f directed edges 6 : E(*D ) —> C, the H irota system consists o f the following equations fo r the function w : V’(D ) —> C, one fo r every quadrilateral face (#o, 2/0, F (V ):

2/i) €

(8.10) 6ow (xo)w (yo) + O iw (y o)w {xi) - 60w (x i)w (y i) - 9 iw (y i)w (x o ) = 0. Note that the Hirota equation coincides with equation (6.30) of Section 6.7 (by the way, this shows that also in that previous version it was natural to assign parameters to directed edges). In terms of the parallelogram real­ ization p : V ( V ) —> C of the quad-graph D corresponding to the labelling 0, equation (8.10) reads: (8.11)

w( xo) w( yo) ( p( yo) - p ( x o)) + w( y0) w ( x i ) ( p ( x i ) - p ( y o ) ) + w ( x i ) w ( y i ) ( p ( y i ) - p ( x i ) ) + w( yi ) w{ xo) ( p{ xo) - p{ yi ) ) = 0.

Obviously, a transformation w i-> cw on V ( S) and w > c~ 1w on V (9 *) with a constant c G C, hereafter called a black-white scaling, maps solutions of the Hirota system into solutions. A relation between the cross-ratio and the Hirota system is based on the following observation:

Theorem 8.6 . (Relation between cross-ratio and Hirota systems) Let w : V ( V ) ( 8 . 12)

C be a solution o f the H irota system. Then the relation z( y)

-

z(x)

=

6 { x , y ) w( x ) w( y )

=

w{ x ) w( y ) ( p ( y )

-

p(x))

316

8. Integrable Circle Patterns

fo r all directed edges (x ,y ) E E ( V ) defines a unique (up to an additive constant) function z : V ( V ) —> C which is a solution o f the cross-ratio system (8.9). Conversely, fo r any solution z o f the cross-ratio system (8.9), relation (8.12) defines a unique (up to a black-white scaling) fu nction w : V (T )) —►C; this function w solves the H irota system (8.10).

In particular, the trivial solution z (x ) = p (x ) of the cross-ratio system corresponds to the trivial solution of the Hirota system, w (x ) = 1 for all x E V (D ). By a direct computation one can establish the following fundamental property.

Theorem 8.7. (Integrability of Hirota system) The Hirota system (8.10) is 3D consistent . As a usual consequence, the Hirota system admits Backlund transforma­ tions and possesses zero curvature representation with transition matrices along the edge ( x,y) E E(T>) given by

(8.13)

*,,«.**> -(’

1 A0/w (x)

w (y )/ w (x )J

where 9 = p (y ) — p (x ).

8.3. Integrable circle patterns Returning to circle patterns, let { z ( x ) : x € V '(S )} be the intersection points of the circles of a pattern, and let { z (y ) : y € V'( S*) } be their centers. Due to (8.1), the function 2 : V(T>) —►C satisfies a cross-ratio system with Q : E ( S) U E (S * ) ►S1 defined as Q (e ) = exp(2 i{e)). Because of (8.2), the first of the integrability conditions (8.8) is fulfilled for an arbitrary circle pattern. Therefore, integrability of the cross-ratio system for circle patterns with prescribed intersection angles : E ( S*) —> (0 ,7r) is equivalent to (8.14)

n

e x p (2 # (e *)) = 1,

Vy0 € V (S * ).

e*Gstar(i/o;S*)

This is equivalent to the existence of the edge labelling a : E ( D ) —> C such that, in the notation of Figure 8.2, (8.15)

exp(2 i*) = — . ai

Moreover, one can assume that the labelling a takes values in S1. Our definition of integrable circle patterns will require somewhat more than integrability of the corresponding cross-ratio system.

317

8.3. Integrable circle patterns

Definition 8.8 . (Integrable circle pattern) A circle pattern with pre­ scribed intersection angles (j) : E (Q *) —> (0,7r) is called integrable if

(8.16)

exp(i(e*)) = 1,

Vy0 <E F (S *),

e*€star(t/0;S*)

i.e., if fo r any circle o f the pattern the sum o f its intersection angles with all neighboring circles vanishes

(mod 2tt).

This requirement is equivalent to a somewhat sharper factorization than (8.15), namely, to the existence of a labelling of the directed edges 0 : E ( D ) —> S1 such that, in the notation of Figure 8.2,

(8.17)

exp (i) =

&

exp (icj)*) =

(O f course, the last condition yields (8.15) with a = 92.) The parallelogram realization p : V(T>) —> C corresponding to the labelling 0 G S1 is actually a rhombic one.

Theorem 8.9. (Isoradial integrability criterion) Combinatorial data S and intersection angles (j) : E ( S) —►(0,7r) belong to an integrable circle pattern if and only i f they admit an isoradial realization. In this case, the dual combinatorial data S* and intersection angles (j) : E ( S*) —> (0 ,7r) admit a realization as an isoradial circle pattern, as well.

Proof. The rhombic realization p : 1^(2)) —►C of the quad-graph D cor­ responds to a circle pattern with the same combinatorics and the same intersection angles as the original one and with all radii equal to 1, and, simultaneously, to an analogous dual circle pattern. □ Consider a rhombic realization p : V ( V ) —> C of D. Solutions z : V'(D ) —> C of the corresponding integrable cross-ratio system which come from integrable circle patterns are characterized by the property that the 2:-image of any quadrilateral (#o, yo, y i) from F ( D ) is a kite with the prescribed angle (j) at the black vertices z (x o), z (x i) (cf. Figure 8.2). It turns out that the description of this class of kite solutions admits a more convenient analytic characterization in terms of the corresponding solutions w : y (!D ) —►C of the Hirota system defined by (8.12).

Theorem 8.10 . (Circle pattern solutions of Hirota system) The so­ lution z o f the cross-ratio system corresponds to a circle pattern if and only if the solution w o f the Hirota system, corresponding to z via (8.12), satisfies the condition

(8.18)

w (x ) € S1,

w (y) € M+,

Vx € V (5 ), y € F (3 *).

318

8. Integrable Circle Patterns

The values w (y) G R+ have then the interpretation o f the radii o f the circles C (y ), while the (arguments o f the) values w (x ) G S1 measure the rotation o f the tangents to the circles intersecting at z (x ) with respect to the isoradial realization o f the pattern.

Proof. As is easily seen, the kite conditions are equivalent to = 1

and

□

This yields (8.18), possibly upon a black-white scaling.

The conditions (8.18) form an admissible reduction of the Hirota system with 9 G S1, in the following sense: if any three of the four points w (x o ), w(yo), w (x i), w (y i) satisfy the condition (8.18), then so does the fourth one. This is immediately seen, if one rewrites the Hirota equation (8.10) in one of the two equivalent forms:

(8 19)

= Qiw(yi) - Qowjyo)

‘

w (x 0)

^

0iw (yo) - 0ow (y i)

w(yi) = %w(x0) + Qiw(xi) w (y0)

0ow {x i) + O iw (xQ) '

As a consequence of this remark, we obtain Backlund transformations for integrable circle patterns.

Theorem 8.11. (Backlund transformations of integrable circle pat­ terns) Let all 9 G S1, and let p : V(T>) —> C be the corresponding rhombic realization o fX ). Let the solution w : V'(D) —> C o f the Hirota system corre­ spond to a circle pattern with combinatorics of S, i-e., satisfy (8.18). Con­ sider its Backlund transformation w+ : V(T>) —> C with an arbitrary param­ eter A

G

S1 and with an arbitrary initial value iv+ (x q)

G

M+ or w+ (yo)

G

S1.

Then

(8.20)

w+ ( x ) e R + ,

w+ { y ) e s \

Vx € V ( 9 ) , y e V{ S* ) ,

so that w+ corresponds to a circle pattern with combinatorics o f S*, which we call a Backlund transform o f the original circle pattern.

We close this section by mentioning several Laplace type equations which can be used to describe integrable circle patterns. First of all, the restriction of the function z to K (9 ) (i*e., to the intersection points of the circles) satisfies the equations n

Here z ( x o) is any intersection point where n circles C ( y \),. . ., C ( y n) meet, z(x k ) is the second intersection point of C (y k ) with C (y k + i) for each A:, and the ak are the labels on the edges ( xo,yk) G E ( V ) . Analogously, the

8.4. za and log 2 circle patterns

319

restriction of the function 2 to V^(9*) (i.e., to the centers of the circles) satisfies the equation m

Here z(yo) is the center of any circle C (y o) that intersects the m circles C ( y i ) , . . . , C (y m) with centers at the points z(yj)\ the intersection of C (y o) with C (y j) consists of two points z ( x j - 1 ), z (x j), and a j are the labels on the edges ( yo, Xj ) G E( T>). These two Laplace type equations follow from the first claim of Theorem 6.31 applied to the cross-ratio system, which is nothing but the case (Ql)<$=o ° f Theorem 6.32. A similar construction can be applied to the Hirota system, written in the three-leg form (8.19). Again, it yields two multiplicative Laplace type equations — on 9 and on 9 *. It is instructive to look at the equation on 9 * (for the radii r j = w (y j) of the circles):

Due to (8.17), this equation can be written in terms of the intersection angles 4>j of C (y o) with C (y j ), and it takes the form of (8.4). Interestingly, the latter equation holds for any circle pattern and is not specific for integrable ones (as opposed to the similar Laplace type equation on 9 )8.4. 2 a a n d l o g z c irc le p a tte r n s Due to the 3D consistency of the cross-ratio and the Hirota systems, we can follow the procedure of Section 6.8 and extend solutions of these systems from a quasicrystallic quad-graph D, realized as a quad-surface C to the whole of Z d (more precisely, to the hull of fi©)- Then, one can ask about isomonodromic solutions. This leads to discrete analogs of the power function. Naturally, these discrete power functions are defined on the same branched covering S of the set Um=i function of Section 7.5.

c ^

as

discrete logarithmic

The discrete cross-ratio system on Z d reads:

(8.21)

q{z( n) , z( n + e,-), z( n + ej + ek), z( n + ek) ) = Oj/Ol,

and possesses the discrete zero curvature representation with transition ma­ trices along the edges (n, n + e*) of Z d given by

320

8. Integrable Circle Patterns

Through the transformation z (n + ek) — z( n) = 0kw( n) w( n + e*),

(8.23)

the solutions of the cross-ratio system are related to the solution of the Hirota system in Z d, (8.24)

9 j w( n ) w( n + ej ) + 0kw (n + e j ) w ( n + ej + e k) —8j w( n + ej + e k) w ( n + e*) — 0kw ( n + ek) w ( n ) = 0.

The latter system possesses a discrete zero curvature representation with transition matrices along the edges (n, n + e k) of Z d given by 1

- 0 kw ( n + e k)

X0k/ w ( n )

w ( n + e k) / w ( n )

(

Special solutions of these two systems on S are defined by the following choice of initial data.

Definition 8.12. (Discrete z 2a) F or a

G

(0,1), the discrete z2a is the

solution o f the cross-ratio system on S defined by the values on the coordinate semiaxes Zn

— z (n e r ), r

( q o£\

G

[ra, m + d —l\, which solve the recurrence relation

i z n+ 1 — z n ) { z n ~ z n —l )

n ------------- —------------ 1 — azn zn+ 1 Zn—1

(8.26)

with the initial conditions z ^ = z(0 ) = 0,

(8.27)

z [r^ = z (e r ) = 02a = exp(2alog0r ),

where logdr is chosen in the interval (7.26).

Definition 8.13. (Discrete tu2a_1) F or a

G

(0,1), the discrete w2a~ l is

the solution o f the Hirota system on S defined by the values on the coordinate semiaxes Wn ^ = w (n e r ), r relation

G

[ra,ra + d — 1], which solve the recurrence

<8 ' 2 8 >

with the initial conditions

(8.29)

= w ( 0) = 0,

= w (e r ) = 02a~ l = exp((2a — 1 ) log0r),

where log 6r is chosen in the interval (7.26).

By induction, one can derive the following explicit expressions for the (r)

solutions Zn .

8.4. za and log 2 circle patterns

321

and for w (r). (8.31,

= n k=l

, = e - ‘.

k —a

Observe the asymptotic relations for n —> oo: (8.32)

4 r> = c(a)(n0r) 2a( l + 0 ( 0 ) ,

(8.33)

= c(a)n2a J( 1 + 0 (n : )).

The main technical advantage of the w variables is seen from the following observation.

Theorem 8.14. (Discrete z 2a defines a circle pattern) The function w2a~ l takes values in R+ at the white points and values in S1 at the black points. Therefore, the function z2a defines a circle pattern.

Proof. The claim for w2a~ l on the coordinate axes is obvious from the explicit formulas (8.31), and can be extended to the whole of S according to the remark after Theorem 8.10. The statement for z2a is now a consequence of Theorem 8.10. □ The restriction of z2a to various quad-surfaces give the discrete analogs of the power function on the corresponding quasicrystallic quadgraphs D with the set 0 = {=t#i ,. . . , ±0 ^} of edge slopes; see Figure 8.4. These pictures lead to the conjecture that the circle patterns z2a are embed­ ded. One possible approach to the analytic study of these patterns could be based on applying the well-developed techniques of the theory of isomonodromic solutions. For either of the systems one can introduce the moving frame as in (7.34): $ (n + e k\A) = L k(n ; A )V (n ; A), and define its logarithmic derivatives as in (7.35):

A(n;A) = - ^

-~ U - 1(n;A).

Theorem 8.15. (Discrete z 2a is isomonodromic) Consider the solution o f the cross-ratio system in (Z + )rf with the initial data (8.30). F or a proper choice o f A (0 ; A), the matrices A (n ; A) at any point n G (Z + )d have simple poles only:

(8.34)

= ^

+

A

/= !

_

322

8. Integrable Circle Patterns

F ig u r e 8.4. Circle patterns z4^5 with combinatorics of the square grid, and z2^3 with combinatorics of the regular hexagonal lattice (isotropic and nonisotropic).

with

/ x (8.35)

(8.36)

, I ~ a/2 A {0\ n ) = 1 '

B {l\ n ) =

—a z ( n ) V ’

~

z ( n + e t) - z ( n - e ()

z ( n + e{ ) - z { n )

( z { n + e t) - z ( n ) ) ( z ( n ) - z ( n - e,))

1

z ( n ) — z ( n — e t)

A t any point n G S, the discrete z2° satisfies the follow ing constraint:

(8.37)

T n , {z{n + ^ ~ Pt 3 z{n + e j ) - z { n - e j )

^ ^

= „ * („ ). y J

8.4. za and log z circle patterns

323

Theorem 8.16. (Discrete w 2a~ x is isomonodromic) Consider the solu­ with the initial data (8.31). F or a proper

tion o f the H irota system in

choice o f A ( 0; A), the matrices A (n ; A) at any point n £ (Z + )d have simple poles only:

„ A ^ (n ) A {n ; A ) - ---- -

(8.38)

A

A # ( n ) h^ ;=i — ei

with

(8.39)

,4(0)(n ) =

(8.40) . _ _________ rn

I w in + e i )

w(n + et) + w(n - et) 1

Oiw{ n + e i ) w { n - e { )

i /gl

w { n - e t)

The upper right entry o f the m atrix A ^ ( n ) ,

denoted by the asterisk in

(8.39), is given by A ^ i n ) — — 5Zf=i ^12 (n )-

A t any point n € S, the

discrete w2a~ l satisfies the following constraint:

4

"

w(n + ei)

w(n- ei) = {a_ h){1_ (_ i r ,+...+ njy

w (n + e i) + w (n — e i)

V

2 /V

J

Proof. The proof of both theorems follows the same scheme as the proof of Theorem 7.14: one first shows that the poles of A (n e r \A) remain simple, due to the recurrence relations (8.26), resp. (8.28), and then shows that the order of poles does not increase at the points n away from the coordinate axes, due to the multidimensional consistency. □ The transition between 2 and w variables is a matter of straightforward computations. Actually, both theorems are dealing with the same matrices but written in different variables. It is interesting to study the limiting behavior of the function z2a as a —> 0. It is not difficult to see that for all n 7^ 0 one has z2a(n ) —> 1 . Denote /« (8.42)

-r / x 1. z2a(n ) - 1 L (n ) = lim -----—----- . a—>o za

This function is called the discrete logarithmic fu nction ; it should not be confused with the namesake function £ (n ) in the linear theory (Section 7.5). Prom (8.42) the following characterization is found: the discrete logarithmic function L is the solution of the discrete cross-ratio system on S defined by

324

8. Integrable Circle Patterns

the values on the coordinate semiaxes L which solve the recurrence relation (3 43)

= L (n e r), r

G

[ra,m + d — 1],

n (^n+1 ~ Lyi)(L n ~ L n- l ) _ 1

Ln+1

Ln—1

2

with the initial conditions (8.44)

L J'1 = L (0 ) = oo,

L 'r) = L (e r ) = log0r,

where log#r is chosen in the interval (7.26). Explicit expressions: (8.45)

= log 6r +

L 2n+1 = log

- + — , fc=l

+ y i j: . fc=l

Theorem 8.17. (Circle pattern logarithm is isomonodromic) The discrete logarithm is isomonodromic and satisfies, at any point n

G

S, the

following constraint: /o

(L (n + ej) - L (n ))(L (n ) - L (n - ej ) ) _ 1

1

}

^

J

L (n + e i ) - L ( n - Cj) 2'

By restriction to quad-surfaces we come to the discrete logarithmic function on arbitrary quasicrystallic quad-graphs D. By construction, they all correspond to circle patterns. A conjecture that these circle patterns are embedded seems plausible (see Figure 8.5).

Figure 8.5. Discrete logarithm circle patterns with combinatorics of the regular square and hexagonal lattices.

8.5. Linearization Let 6 : E(T>) —» C be an edge labelling, and let p : V(T>) —> C be the corresponding parallelogram realization of D defined by p (y ) —p (x ) = 0 ( x , y ). Consider the trivial solutions zq

(

x

)

= p (x ),

w q

(

x

)

= 1,

Vx € V (D )

325

8.5. Linearization

of the cross-ratio system (8.9) and the corresponding Hirota system (8.11). Suppose that zq : V(T>) —> C belongs to a differentiable one-parameter family of solutions : V(T>) —> C, e G ( —eo,eo), of the same cross-ratio system, and denote by we : V(T>) —> C the corresponding solutions of the Hirota system. Denote (8.47) T h e o re m 8.18. (D is cre te d eriv a tiv e for d iscrete holom orphic func­ tio n s) Both functions f , g : V ( V ) —> C solve discrete Cauchy-Riemann equations (7.16). P r o o f. By differentiating (8.12), we obtain a relation between the functions f,g :V ( D )-*C : (8.48)

g (y ) ~ g (x ) = ( f ( x ) + f { y ) ) ( p ( y ) - p ( x ) ) ,

V (x ,y ) € E ( D).

The proof of the theorem is based on this relation solely. Indeed, the ex­ actness condition for the form on the right-hand side on an elementary quadrilateral reads ( f ( x o) + f( yo) ) (p(yo) - p (x 0)) + (f( yo) + f ( x i ) ) (p(xi) - p(yo)) + { f ( x l) + f ( y i ) ) (p (y i) - p (x i)) + ( f ( y i ) + f { x 0)) (p(a:0) - p (y i)) = 0,

which is equivalent to (7.16) for the function /. Similarly, the exactness condition for /, that is, ( f ( x o) + f{y o )) - ( f (y o ) + f i x 1 )) + { f ( x i) + f ( y i ) ) - ( f ( y i ) + f ( x 0) ) = 0,

yields g(yo ) ~ g {x o) _ g ( x i ) - g ( y o )

g ( y i ) - g ( x i ) _ g (x 0) - g { y i ) = Q

p ( y o ) - p ( x 0)

p { y i)-p { x i)

p (x i)-p (y o )

p (x o )-p (y i)

'

Under the condition p(yo) —p {x o) = p (x i) —p (y i), this is equivalent to (7.16) for g.

□

R em ark . This proof shows that, given a discrete holomorphic function / : V(T>) —►C, relation (8.48) correctly defines a unique, up to an additive constant, function g : V (D ) —> C, which is also discrete holomorphic. Con­ versely, for any g satisfying the discrete Cauchy-Riemann equations (7.16), relation (8.48) defines a function / uniquely (up to an additive black-white constant); this function / also solves the discrete Cauchy-Riemann equa­ tions (7.16). Actually, formula (8.48) expresses that the discrete holomor­ phic function / is the discrete derivative of g , and so g is obtained from / by discrete integration. Summarizing, we have the following statement.

326

8. Integrable Circle Patterns

Theorem 8.19. (Linearization of circle patterns) a) A tangent space to the set o f solutions o f an integrable cross-ratio system, at a point corresponding to a rhombic embedding o f a quad-graph, consists o f discrete holom orphic functions on this embedding. This holds in both descriptions o f the above set: in terms o f variables z satisfying the cross­ ratio equations, and in terms o f variables w satisfying the Hirota equations. The corresponding two descriptions o f the tangent space are related via the discrete derivative (resp. antiderivative) o f discrete holom orphic functions.

b) A tangent space to the set o f integrable circle patterns o f a given combinatorics, at a point corresponding to an isoradial pattern, consists o f discrete holomorphic functions on the rhombic embedding o f the correspond­ ing quad-graph, which take real values at white vertices and pure imaginary values at black ones. This holds in the description o f circle patterns in terms of circle radii and rotation angles at intersection points (H irota system).

A spectacular example of this linearization property is delivered by the isomonodromic discrete logarithm studied in Section 7.5 and isomonodromic z2a circle patterns of Section 8.4.

Theorem 8.20. (Linearization of w 2a~ x circle patterns is the dis­ crete logarithm) The tangent vector to the space o f integrable circle pat­ terns along the curve consisting o f patterns w2a~ l , at the isoradial point corresponding to a = 1/2, is the discrete logarithmic fu nction i defined in Section 7.5.

Proof. We have to prove that the discrete logarithm £ and the discrete power function w2a~ l are related by

Due to Theorem 8.18, it is enough to prove this for the initial data on the coordinate semiaxes. But this follows by differentiating with respect to a the initial values (8.31) at the point a — 1/2, where all w — 1: the result coincides with (7.31).

□

8 .6 . Exercises 8.1. Check that formulas (8.30), (8.31) give solutions to the corresponding difference equations (8.26), (8.28). 8.2. Prove asymptotic relations (8.32), (8.33).

8.3. Fill in the details of the proofs of Theorems 8.15, 8.16.

327

8.7. Bibliographical notes

8.4. For every solution 2 : Z d —> C of the cross-ratio system (8.21), define the dual solution 2 * : Z d —* C by 92 z * ( n + e j ) - z * ( n ) = — -------- { ----v Jl v ' z(n + ej) — z(n)

The dual solution is defined uniquely up to translation, and this freedom can be fixed by prescribing 2*(0). Show that for a G (0,1) the dual solution to the discrete z2a, normalized to vanish at n = 0, coincides with the discrete z2{x~a\

8.5. Show that the limit a —■> 1 in Definition 8.12 leads to the discrete z 2 as a solution of the cross-ratio equation, satisfying the recurrence relations (8.26) with a = 1 on the coordinate semiaxes, and with the initial data

2<0)=0,

z( e1) = 0 ,

z( 2e, ) =0>.

z(e, + e„) =

■

In particular, one sector of the discrete z 2, defined on (Z + ) 2, in the case of 9\ = 1, 92 = i, is characterized by the initial data 2 ( 0 , 0 ) = 2 ( 1 , 0 ) = 2 ( 0 ,1) = 0 ,

2 ( 2 , 0 ) = 1,

2 ( 0 , 2 ) = -1 ,

z(l,l) = i - . 7r

8.6. Show that the dual solution to the discrete z2 is the discrete logarithm L. 8.7. Show that for the cross-ratio system on (Z + )2 with 9\ = 1, 9^ — i , the dual solution to z (m , n ) — l/ (m + in ) is given by z * (m , n ) = - ((m + in )3 — (m — in )). o

This can be regarded as the discrete z3.

8.7. Bibliographical notes Section 8.1: C ircle patterns. The idea that circle packings and, more generally, circle patterns serve as a discrete counterpart of analytic functions is by now well established; see the monograph by Stephenson (2005). The origin of this idea is connected with the approach by Thurston (1985) to the Riemann mapping theorem via circle packings. Since then the theory bifurcated to several areas. One of them is dealing mainly with approximation problems. The most popular are hexagonal packings, for which the convergence to the Riemann mapping was established in Rodin-Sullivan (1987). In He-Schramm (1998) it was shown that this convergence actually holds in the class C°°, that is, all higher derivatives are approximated. Similar results are available also for

328

8. Integrable Circle Patterns

circle patterns with combinatorics of the square grid introduced in Schramm (1997), and even for more general circle patterns; see Bucking (2007). Another area concentrates around the uniformization theorem of KoebeAndreev-Thurston, and is dealing with circle packing realizations of cell com­ plexes of prescribed combinatorics, rigidity properties, constructing hyper­ bolic 3-manifolds, etc.; see Thurston (1997), He (1999), Stephenson (2005). A variational description of circle packings was initiated by Colin de Verdiere (1991). Further progress is due to Bragger (1992), Rivin (1994), and Bobenko-Springborn (2004). The extremals of the functional used in the last paper are described by equation (8.4). An application of this approach in discrete differential geometry is the construction of discrete minimal surfaces through circle patterns in Bobenko-Hoffmann-Springborn (2006). The main topic of this chapter is interrelations of circle patterns with integrable systems. See the notes to Section 8.3.

Section 8.2 : Integrable cross-ratio and Hirota systems. In this gen­ erality (for arbitrary quad-graphs) this material is due to Bobenko-Suris (2002a). On 1? the relation between the cross-ratio and Hirota systems is considered in Capel-Nijhoff (1995). Our presentation follows BobenkoMercat-Suris (2005).

Section 8.3: Integrable circle patterns. Orthogonal circle patterns with combinatorics of the square grid were studied in Schramm (1997). Hexagonal circle patterns with fixed intersection angles were investigated in BobenkoHoffmann (2003), and with the multiratio property, in Bobenko-HoffmannSuris (2002). The general theory presented here is formulated in BobenkoMercat-Suris (2005). Section 8.4: z a and log z circle patterns. The circle patterns za on the square lattice were introduced in Bobenko (1999) and studied in BobenkoPinkall (1999) and Agafonov-Bobenko (2000). The conjecture that these patterns are embedded, i.e., the interiors of different kites are disjoint, was formulated in the first of these papers. The study was extended to the reg­ ular hexagonal grid in Bobenko-Hoffmann (2003). The fact that the circle patterns za are immersed, i.e., the neighboring kites do not overlap, was proven in Agafonov-Bobenko (2000) for the square grid and in AgafonovBobenko (2003) for the hexagonal grid combinatorics. The embeddedness was proven in Agafonov (2003) for the case of the square grid combina­ torics. The isomonodromic constraint (8.37) was obtained first for a = 1/2 in Nijhoff (1996), with no geometric interpretation. For the Hirota sys­ tem, the isomonodromic constraint (8.41) was derived in Nijhoff-RamaniGrammaticos-Ohta (2001), also with no relation to geometry. Our presen­ tation here follows Bobenko-Mercat-Suris (2005).

8.7. Bibliographical notes

329

Section 8.5: Linearization. The operation of discrete integration for discrete holomorphic functions was considered in Duffin (1956, 1968), and Mercat (2001). Linearization of circle patterns was studied in BobenkoMercat-Suris (2005); in particular, the derivation of Green’s function from the za circle pattern is taken from this paper. Section 8 .6 : Exercises. Ex. 8.3: See Bobenko-Mercat-Suris (2005). Ex. 8.5: See Agafonov-Bobenko (2000). Ex. 8.6: See Agafonov-Bobenko (2000) in the case of the regular square grid. Ex. 8.7: See Bobenko-Pinkall (1999).

Chapter 9

Foundations

For the reader’s convenience we give here a brief introduction to projective geometry and the geometries of Lie, Mobius, Laguerre and Pliicker. We also include a number of classical incidence theorems relevant to discrete differential geometry. For extensive presentations of these classical results we recommend, in particular, the textbooks: Blaschke (1954) and Pedoe (1970) on projective and Pliicker line geometry, Blaschke (1929) on sphere geometries, Cecil (1992) on Lie geometry, and Hertrich-Jeromin (2003) on Mobius geometry.

9.1. Projective geom etry Projective geometry studies properties of geometric objects which remain invariant under the group of projective transformations, which is generated by Euclidean motions, homotheties, and central projections. A suitable analytical framework for doing projective geometry is given by the notion of homogeneous coordinates. The main space of real projective geometry is R P ^ = P (R 7V+1), which is the set of equivalence classes of M'v+I \ {0 } modulo the following equivalence relation: x ~ y

x = Xy,

x, y G R JV+1 \ { 0} ,

A € R*.

On a general note, building the set of equivalence classes with respect to the relation ~ is called a projectivization. Thus, points of P R N are projectivizations of 1-dimensional vector subspaces of MjV+1. 331

332

9. Foundations

The equivalence class of x = ( x i , . . ., x#, % n+ i) £ K iV+1 \ {0 } is denoted by [x] — [x\ : • • • : x ^ • £jv+i]- The space R N+1 is called the space of homogeneous coordinates on WPN . One says that x G R N+1 \ {0 } is a lift of [x] to the space of homogeneous coordinates, or a representative of [x] in the space of homogeneous coordinates. The usual space R N can be identified with the subset of equivalence classes of elements of R N + l with xjv+i 7^ 0: x = ( x i , . .., Xat)

G

RN ^

[x\ : • • • : X]y : 1]

G

PR ^.

This subset is called an affine part of PR ^. The complement of an affine part, i.e., the set of equivalence classes [xi : • • • : x/v • 0], is called the hyperplane at infinity , and its elements are called infinitely remote points. Of course, x ^ + i plays a distinguished role in this construction. One can single out a coordinate other than xat+1 and will then obtain different affine parts. The N + 1 affine parts obtained in this way build an atlas of P R ^ as a real manifold, consisting of N + 1 charts. More generally, projective hyperplanes in P R ^ are projectivizations of hyperplanes, that is, of vector iV-spaces in R N+1. Any hyperplane u can be described by an equation N+ 1 (ifc, X ) =

U iX i =

i= 1

where (ixi,... , u n , u n + i ) £ ( R ^ 1)* \ {0 }, and (•,•) denotes the pairing between the dual spaces R N+1 and (R ^ 4"1)*. Actually, only the equivalence class [u\ : • • • : u m : 'Mjv+i] is relevant in this description, and a hyperplane can be identified with this equivalence class. One calls [u\ : • • • : : u/v+i] the homogeneous coordinates of a hyperplane u. For instance, the hyperplane at infinity has homogeneous coordinates [0 : • • • : 0 : 1]. Thus, the set (R P ^ )* of all projective hyperplanes is isomorphic to R P N , again. Interchanging the roles of points from R P ^ with hyperplanes from (R P ^ )* is the projective duality.

For any 1 < d < N — 1, a projective d-space in R P ^ is a projectivization of a vector (d + l)-space in R N+1. There are two dual ways to describe a projective d-space. • Let x i , . . . , x ^+1 be d + 1 points of WPN in general position with representatives x i , ... , x^+i in the space of homogeneous coordi­ nates. The general position condition means that the vector space E = span(xi,... ,Xrf_(_i) has dimension d + 1. Then P ( £ ) is the d-dimensional space through x i , . . . , Xd+\• The points of P ( £ ) are given, in homogeneous coordinates, by all possible linear combina­ tions x = cixi H------ b cd+i x d+ i with ( c i , . . . , cd+ 1 ) ^ ( 0 , .. . , 0).

333

9.1. Projective geometry

• Alternatively, let u \,. . ., u ^ -d be N — d hyperplanes of (WP^)* in general position, with representatives u\y. . . ,U N -d in the space of homogeneous coordinates. Again, the general position condition means that the vector space £-*- = span(tii,... , nyy-d) has dimen­ sion N — d. Then the vector space S = { x G R n+1 : (&i,x) = • • • = (iiN -d ,x ) — 0} has dimension d + 1, and P (E ) is a projective d-space defined as the intersection of the hyperplanes u i , . . . , 'M/v-dIf d\ + c?2 > N , then the intersection of a di-space with a c?2"Space in MFN is a projective space of dimension > d\ + d
a line a point a point on a line the line connecting two points three points are collinear

In the three-dimensional projective geometry, the incomplete vocabulary looks as follows: a point a line a plane a line through a point the intersection line of two planes the intersection point of a line with a plane the intersection point of three planes four planes have a common point

a plane a line a point a line in a plane the line connecting two points the plane through a line and a point the plane through three points four points are coplanar

334

9. Foundations

Projective transformations or collineations of MFN are induced by non­ degenerate linear maps on the space of homogeneous coordinates: y = Tx, r g g l ( n + i,m).

Theorem 9.1. (Fundamental theorem of projective geometry) a,) Let 7 : MFN —>R P ^ be an injective map such that 7 ( R P N ) does not

lie in a hyperplane, and fo r any three collinear points X\,X 2 ,X 3 their images 7 (# i), 7 ( 2:2), 7 (^ 3) are a^so collinear. Then 7 is a projective transformation. b) F or any two sets { # 1 , . .. , £ ; v + 2 } C R P N and { y\, . . . , 2/7V + 2 } C R P N such that in each set no N + 1 points lie in a hyperplane, there is a unique projective transformation 7 such that yk = 7 ( 2:^) /or all k = 1, .. ., N + 2.

A projective transformation of a line is characterized by the property of preserving the cross-ratios of four points. Finally, we briefly discuss the notion of quadric in a projective space. Let Q : R ^ 1 x R N+1 —>M be a nondegenerate symmetric bilinear form; we will denote the matrix of this form by Q. The set of points x G M FN with homogeneous coordinates x G R N + 1 satisfying the quadratic equation N+i (9.1)

Q (x ,x ) =

^ 2 Q j kXj Xk = 0 j,k= 1

is called a (nondegenerate) quadric Q C WPN . Of course, only those nonempty quadrics are interesting which correspond to indefinite bilinear forms. In particular, a nondegenerate quadric in MP2 is called a conic. Two points x, y G R P ^ with homogeneous coordinates x ,y G R N+1 are called conjugate with respect to a quadric if N+l

Q ( x , y) =

X I

Qj kXj V k = 0.

j,k=1

The points conjugate to a given point x G WPN build the polar hyperplane of x. Thus, the polar hyperplane is defined as P (x _L), where the orthogonal complement is taken according to the scalar product Q : X1 = {y € R n+1 : Q { x ,y ) = 0 }.

Homogeneous coordinates of the polar hyperplane can be chosen as u = Qx.

Thus, the polar point x of a hyperplane u has homogeneous coordinates x = Q l u.

335

9.2. Lie geometry

Two hyperplanes u, ^ with homogeneous coordinates ii,v E R ^ -1-1 are called conjugate with respect to the quadric Q if N+1

^ ^ (Q

^jkUjVk — 0 *

j,k= 1

Each of them contains the polar point of the other. A tangent hyperplane to the quadric Q is self-conjugate, so its homogeneous coordinates satisfy the quadratic equation N+l

(9.2)

^ 2 (Q ~ l ) j kUj Uk = 0. j , k= 1

A quadric can be viewed either as the set of points satisfying (9.1) or as the envelope of its tangent hyperplanes satisfying (9.2). The polarity relation can be generalized from points and hyperplanes to projective spaces of arbitrary dimensions. For a projective d-space U = F (U ) one defines a polar subspace as P([/-L), where the orthogonal complement is understood with respect to the scalar product Q. Polarity can be regarded as a generalization of duality.

9.2- Lie geom etry 9.2.1. O b je cts o f L ie g eo m etry . The following geometric objects in the Euclidean space R N are elements of Lie geometry: • Oriented hyperspheres. A hypersphere in R ^ with center c G R ^ and radius r > 0 is described as S = { x G R ^ : \x — c\2 = r2}. It divides R ^ into two parts, inner and outer. Declaring one of the two parts of R ^ to be positive, we come to the notion of an oriented hypersphere. Thus, there are two oriented hyperspheres S ± for any 5. One can take the orientation of a hypersphere into account by assigning a signed radius ± r to it. For instance, one can assign positive radii r > 0 to the hyperspheres with the inward field of unit normals and negative radii r < 0 to the hyperspheres with the outward field of unit normals. • Oriented hyperplanes. A hyperplane in R ^ is given by the equation P — { x G R n : (v ,x ) = d}, with a unit normal v G SN ~ 1 and d G R. Clearly, the pairs (v, d) and ( —v, —d) represent one and the same hyper plane. It divides R ^ into two half-spaces. Declaring one of the two half-spaces to be positive, we arrive at the notion of an oriented hyperplane. Thus, there are two oriented hyperplanes P ± for any P . One can take the orientation of a hyperplane into

336

9. Foundations

account by assigning the pair (v, d) to the hyperplane with the unit normal v pointing into the positive half-space. • Points. One considers points x G R N as hyperspheres of a vanishing radius. • Infinity. One compactifies the space R N by adding the point at infinity oo, with the understanding that a basis of open neighbor­ hoods of oo is given, e.g., by the outer parts of the hyperspheres \x\2 = r 2. Topologically the compactification so defined is equiva­ lent to a sphere S^. • Contact elements. A contact element is a pair consisting of a point x G R n and an (oriented) hyperplane P through x; alternatively, one can use a normal vector v to P at x. A contact element rep­ resents the (equivalence class of) hypersurfaces through the point x with the tangent hyperplane P at x. In the framework of Lie geometry, a contact element can be identified with a set (a pencil) of all hyperspheres S through x which are in oriented contact with P (and with one another), thus sharing the normal vector v at x; see Figure 9.1.

F i g u r e 9.1. Contact element.

9.2.2. P r o je c t iv e m od el o f L ie geom etry. All the above elements are modelled in Lie geometry as points, resp. lines, in the ( N + 2)-dimensional projective space P(MiV+1’2) with the space of homogeneous coordinates jgJV+1,2^ rpkg }a^ er js Space spanned by the N + 3 linearly indepen­ dent vectors e i , . . . , e j v +3 and equipped with the pseudo-Euclidean scalar

337

9.2. Lie geometry

product 1, —1,

i = j G { 1 , . . . , N + 1}, i = j G { N + 2, N + 3},

0,

i ^ j.

It is convenient to introduce two isotropic vectors (9.3)

eo = ^(e7v+2 - ©ah-i),

&oo = 5 ( ^ + 2 + e N + 1 ),

for which (e 0, eo) — (^ 005^00) — O,

2*

®oo)

The models of the above elements in the space K ^ +1»2 of homogeneous coordinates are as follows: • Oriented hypersphere with center c

G

M>N and signed radius

r G l:

s = c + e 0 + (|c|2 - r 2)eoo + re/v+3.

(9.4)

• Oriented hyperplane (v ,x ) — d with v

G

and d

G M:

p = v + 0 • e 0 + 2de00 + ejv+ 3 *

(9.5) • P o in t x

G

M.N :

(9.6)

x = x + eo + |x|2eoo + 0 • e ^ +3.

• Infinity oo; (9.7)

do = eoo. • Contact element (x, P ) :

(9.8)

span(i,p).

In the projective space P(MiV+1,2) the first four types of elements are repre­ sented by the points which are equivalence classes of (9.4)-(9.7) with respect to the relation £ ~ rj <=> £ = \ r ) with A G K* for £, 77 G M^-1-1,2. A contact element is represented by the line in P(MiV’1"1,2) through the points with rep­ resentatives x and p. We mention several fundamentally important features of this model: (i) A ll the above elements belong to the Lie quadric PQL^-1"1,2), where (9.9)

L " +1’2 - {£

G

R N + l'2 : (£,£) = 0}.

Moreover, the points of P (L Ar+1,2) are in a one-to-one correspon­ dence with the oriented hyperspheres in R N , including degenerate cases: proper hyperspheres in correspond to points of P (L iV+1,2) with both eo- and e 7v+ 3-components nonvanishing, hyperplanes in R n correspond to points of P (L iV_h1,2) whose eo-component van­ ishes, points in R N correspond to points of P (L ^ + 1,2) with vanish­ ing e/v+3-component, and infinity corresponds to the only point of P (L 7V'f1,2) with both eo- and e;v+ 3-coinponents vanishing.

338

9. Foundations

(ii) Two oriented hyperspheres S\,S 2 are in oriented contact (i.e., are tangent to each other with the unit normals at tangency pointing in the same direction) if and only if |ci - c-212 = (r\ - r2) 2,

(9.10)

and this is equivalent to (§ [,§ 2 ) = 0 . (iii) An oriented hypersphere S = { x G R N : \x —c\2 = r 2} is in oriented contact with an oriented hyperplane P = { x E : (v ,x ) = d} if and only if (9.11)

(c, v) - r — d = 0. Indeed, equation of the hyperplane P tangent to S at xo G S' reads: (xo — c, x — c) — r 2. Denoting by v = (c — x q )/ t the unit normal vector of P (recall that the positive radii are assigned to spheres with inward unit normals), we can write the above equation as (v ,x ) = d with d = (c, (c — x o )/ r) — r — (c ,v ) — r, which proves (9.11). Now, the latter equation is equivalent to (s ,p ) = 0.

(iv) A point x can be considered as a hypersphere of radius r — 0 (in this case the two oriented hyperspheres coincide). A 11 incidence relation x G S with a hypersphere S (resp. x G P with a hyperplane P ) can be interpreted as a particular case of oriented contact of a sphere of radius r — 0 with S (resp. with P ), and it takes place if and only if (x ,s ) = 0 (resp. (x,p) = 0). (v) For any hyperplane P , (60, p) = 0. One can interpret hyperplanes as hyperspheres (of an infinite radius) through 00. More precisely, a hyperplane (v, x ) — d can be interpreted as a limit, as r —> 00, of the hyperspheres of radii r with centers located at c = rv + ?i, with (v ,u ) = d. Indeed, the representatives (9.4) of such spheres are s

=

(rv + u) + e 0 + (2 dr + (w, u ))e 00 + r e N+3

~

(v + 0 ( l / r ) ) + (l/r)eo + (2d + O (l/ r ))e 00 + eyv+3

=

p + 0 (l/ r ).

Moreover, for similar reasons, the infinity 00 can be considered as a limiting position of any sequence of points x with |x| —►00. (vi) Any two hyperspheres S i, S 2 in oriented contact determine a con­ tact element (their point of contact and their common tangent hyperplane). For their representatives 51 , s2 in the line in P(M7V_h1,2) through the corresponding points in P (L 7V+1,2) is isotropic , i.e., lies entirely on the Lie quadric P (L 7V+1,2). This fol­ lows from (QlSi + a 2S250!iSi +a252) = 2aiQ2(5i,S2) = 0 .

9.2. Lie geometry

339

Such a line contains exactly one point whose representative x has vanishing e 7v+ 3-component (and corresponds to x, the common point of contact of all the hyper spheres), and, if x ^ oo, exactly one point whose representative p has vanishing eo-component (and corresponds to P , the common tangent hyperplane of all the hyper­ spheres). In case when an isotropic line contains oo, all its points represent parallel hyperplanes, which constitute a contact element through oo. Thus, if one considers hyperplanes as hyperspheres of infinite radii, and points as hyperspheres of vanishing radii, then one can conclude that: ► Oriented hyperspheres are in a one-to-one correspondence with points of the Lie quadric P (L 7V+1,2) in the projective space P(R iV+1’2). ► Oriented contact of two oriented hyperspheres corresponds to or­ thogonality of (any) representatives of the corresponding points in P(MAr+1’2). ► Contact elements of hypersurfaces are in a one-to-one correspon­ dence with isotropic lines in P(R 7V+1,2). We will denote the set of all such lines by £ ^ +1,2.

9.2.3. Lie sphere transformations. According to F. Klein’s Erlangen Program, Lie geometry is the study of properties of transformations which map oriented hyperspheres (including points and hyperplanes) to oriented hyperspheres and, moreover, preserve the oriented contact of hypersphere pairs. In the projective model described above, Lie geometry is the study of projective transformations of P (R ^ +1,2) which leave P (L Ar+1,2) invariant, and, moreover, preserve orthogonality of points of P (L ^ +1,2) (which is un­ derstood as orthogonality of their lifts to L iV+1,2 C R ^ " 1,2; clearly, this relation does not depend on the choice of lifts). Such transformations are called Lie sphere transformations. Theorem 9.2. (Fundamental theorem of Lie geometry) a) The group o f Lie sphere transformations is isomorphic to the fa ctor 0 ( N + 1,2)/{± / }. b) Every line preserving diffeomorphism o f P (L ^ +1,2) is the restriction to P (L j/v+1,2) o f a Lie sphere transformation.

Since vanishing of the eo- or e 7v+ 3-component of a point in P (L Ar+1,2) is not invariant under a general Lie sphere transformation, there is no distinc­ tion between oriented hyperspheres, oriented hyperplanes and points in Lie geometry.

340

9. Foundations

9.2.4. Planar families of spheres; Dupin cyclides. Considerations of this subsection hold for the geometrically most significant case N = 3. Definition 9.3. (Planar family of spheres) A planar fam ily o f (oriented) spheres in M3 is a set o f spheres whose representatives s G P (L 4,2) are con­ tained in a projective plane P (E ), where £ is a three-dimensional vector subspace o/R 4,2 such that the restriction o f (•, •) to £ is nondegenerate.

Thus, a planar family of spheres is a conic section P (£ fl L 4,2). Clearly, there are two possibilities: (a) The signature of (•, -)|s is (2,1), and so the signature of (•, -)\^± is also ( 2 , 1 ). (b) The signature of (•, *)|s is (1,2), and so the signature of (•, * ) | i s (3,0). It is easy to see that a planar family is a one-parameter family, parametrized by a circle S1. Indeed, if e\,e2,e 3 is an orthogonal basis of £ such that (ei ,ei) = (e2,e 2) = —(^3 , 63) = 1 (say), then the spheres of the planar family come from the linear combinations s = a\e\ + a 2e2 + e3 with (a\ei + a 2e2 + e3, a\e\ + a 2e2 + e3) = 0

&

a\ + a 2 = 1.

In the second case mentioned above, the space E -1 has only a trivial inter­ section with L 4,2, so the spheres of the planar family P (L 4,2 fl £ ) have no common touching spheres.

Definition 9.4. (Cyclidic family of spheres) A planar fam ily o f spheres is called cyclidic if the signature o f (*, *)|s is ( 2 , 1 ), so that the signature o f

(t)Ie-l

is also (2, i).

For any cyclidic family P (L 4,2 fl £ ) there is a dual cyclidic family P (L 4,2 fl E -1) such that any sphere of the first family is in oriented contact with any sphere of the second. The family P (L 4,2 flE), as any one-parameter family of spheres, envelopes a canal surface in M3, and this surface is an en­ velope of the dual family P (L 4,2 fl E*1), as well. Such surfaces are called Dupin cyclides.

Examples: a) Points of a circle build a planar cyclidic family of spheres (of radius zero). The dual family consists of all (oriented) spheres through this circle, with centers lying on the line through the center of the circle orthogonal to its plane; see Figure 9.2, left. The corresponding Dupin cyclide is the circle itself. It can be shown that any Dupin cyclide is an image of this case under a Lie sphere transformation. b) Planes tangent to a cone of revolution build a planar cyclidic family of spheres, as well. The dual family consists of all (oriented) spheres tangent

341

9.3. M obius geometry

to the cone, with centers on the axis of the cone; see Figure 9.2, right. The corresponding Dupin cyclide is the cone itself.

F i g u r e 9.2. Left: A cyclidic family of spheres through a circle. Right: A cyclidic family of spheres tangent to a cone.

9.3. M obiu s geom etry 9.3.1. O b jects o f M ob iu s g eo m etry . Mobius geometry is a subgeometry of Lie geometry, with points distinguishable among all hyperspheres as those of radius zero. Thus, Mobius geometry studies properties of hyper spheres in invariant under the subgroup of Lie sphere transformations preserving the set of points. The following geometric objects are elements of Mobius geometry of : • Points x

G

RN.

• Infinity oo which compactifies • (N onoriented) hyperspheres S — centers c G R N and radii r > 0.

into S^. {x

• (N onoriented) hyperplanes P = { x normals v G SN ~ l and d G M.

: \x — c\2 = r 2} with

G

G

R N : (v ,x ) = d}, with unit

The Mobius group M 6 b (N ) o f M consists of point transformations gen­ erated by reflections in hyperplanes P = { x G R N : (v ,x ) — d}: (9.12)

^ (v, x ) — d x i—> x — 2 — ----- -— v , {v ,v )

and by inversions in hyperspheres S = { x (9.13)

X

G

: \x — c\2 = r 2}:

r2 •—►C -j- --------rx (x — c). \x — c\z

Clearly, Mob(A^) contains as a subgroup the group E ( N ) of Euclidean mo­ tions of R N , which is generated by reflections in hyperplanes. It contains

342

9. Foundations

also dilations, since they can be represented as compositions of inversions in two concentric hyperspheres. For N > 3, the Liouville theorem says that M o b (N ) coincides with the group of conformal diffeomorphisms. Voo

F i g u r e 9.3. Stereographic projection.

One compactifies R N by adding the point oo, thus arriving at the N sphere S^. It is convenient to model as embedded in M.N+1: S N = { y e R N+ 1 : { y, y) = l }

(we use one and the same notation for the scalar products in K'V and in The (inverse) stereographic projection a : R N —> § w \ {yoo} from the north pole Voo — e N + i is defined by

R n+1- its meaning in each case should be clear from the context).

(9.14)

y = ff(x ) = _

2 _

Ixl2 — 1 x + _ _ ejv+i;

see Figure 9.3. The formula <j ( x )

2 2

= eyv+1 +

R

+ 1

(x

2 - e jv + i) = e;v+i + ,------------- ( x

-

e jv + i)

F - ejv+i|

shows that one can view the stereographic projection a also as the re­ striction to R n of the inversion of R N + l in the hypersphere with center ew+i and radius \/2. Setting SN . Hyperplanes and hyperspheres in R N are mapped by the stereographic projection a to hyperspheres in S^, the images of hy­ perplanes being hyperspheres through y ^ . Thus, hyperplanes in R N can be interpreted as hyperspheres through oo.

343

9.3. Mobius geometry

Elements of Mobius geometry of

are:

• Points y e SN . • (NonorientedJ hyperspheres S

C

S^.

Any hypersphere S C §>N , except for great ones, may be described as the intersection of with an affine hyperplane {y G R ^ 4"1 : (s,y) = 1}. The point s G R ^ 1, which is the pole of this hyperplane with respect to §N , lies outside of S^, and S C is the contact set of with the tangent cone to with apex s. Also, S c S N is the intersection of and the orthogonal iV-sphere S C with center s and radius p such that p2 — (s ,s ) — 1, see Figure 9.4. (For a great hypersphere S C S^, which is the intersection of with a hyperplane {y G R ^ 1 : (s ,y ) = 0}, the latter hyperplane also plays the role of the orthogonal TV-sphere S, and the tangent cone becomes a cylinder.)

F i g u r e 9.4. Hypersphere S C §>N and the corresponding point 5 G Mn + 1 , with an orthogonal Af-sphere S through S.

The Mobius group M o b (N ) o f SN is generated by inversions in hyper­ spheres S C given by (9.15)

p2

y i-> s + | ^ (y - s). Iy - s\z

Transformation (9.15) coincides with the restriction to of R 7V+1 in the 7V-sphere S, which is orthogonal to along the hypersphere S.

of the inversion and intersects

A hypersphere S in R ^ (or in SN ) can also be interpreted as the set of points x G S. This allows us to introduce lower-dimensional spheres:

344

9. Foundations

• Spheres. A /c-sphere is a (generic) intersection of N —k hyperspheres Si (i = l , . . . , N - k ) .

They are further objects of Mobius geometry (in contrast to Lie geometry). This means that the class of ^-spheres is preserved by Mobius transforma­ tions. 9.3.2. P r o je c tiv e m od el o f M ob iu s g eo m e try . In modelling elements of Mobius geometry (of either of the spaces R ^ U l o o } or §N ), one can use the Lie-geometric description and just omit the ejv-f 3-component. The resulting objects are points of the ( N + l)-dimensional projective space P(M7V+1’1) with the space of homogeneous coordinates R N+1^. The latter is the space spanned by N + 2 linearly independent vectors e i , . . . , e^+2 and equipped with the Minkowski scalar product f

1? i

(e u e j) = < -1 , {

— j G { 1? • • • ? N +

1}?

i = j = TV+ 2,

0,

i ± j.

We continue to use notation (9.3) in the context of Mobius geometry. Elements of Mobius geometry of R N are modelled in the space M ^ 1,1 of homogeneous coordinates as follows: « P o in t x

G

RN :

(9.16)

x = xeuc = x + e0 + |x|2eoo.

• Infinity oo: (9.17)

oo = eoo• Hypersphere with center c

(9.18)

G

s = 5Euc = c + e0 + (|c|2 - r 2)eoo.

• Hyperplane (v ,x ) = d with v (9.19)

1SLN and radius r > 0:

G

SN ~ l and d G M:

p = pEuc — v + 0 • eo + 2deoc.

In the projective space P(MAr'f1’1) these elements are represented by points which are equivalence classes of (9.16)-(9.19) with respect to the usual re­ lation £ ~ rj £ = A77 with A g K * for f , 77 G Fundamental features of these identifications are the following: (i) The infinity do can be considered as a limit of any sequence of x for x G R n with |x| —►00. The points x G R N U {00} are in a one-toone correspondence with the points of the projectivized light cone P (L Ar+1’1), that is, with the straight line generators of ( 9 .20)

L N + h l = {£ G

: (£,£) =

0 }.

345

9.3. M obius geometry

The points x G R ^ correspond to the points of P (L j/v+1,1) with a nonvanishing eo-component, while oo corresponds to the only point of P (L j/v+1,1) with the vanishing eo-component. Euclidean representatives (9.16) have an important property: (9.21)

(xi, X2 ) = —\\xi ~ 2^2 12>

Vxi, X2

G

RN.

(ii) Hyperspheres s and hyperplanes p belong to P ( R ^ 1,:L), where (9.22)

= {C € R JV+1,1 : (£, £> > 0 } is the set of space-like vectors of the Minkowski space R ^ +1,1. Hy­ perplanes can be interpreted as hyperspheres (of an infinite radius) through oo.

(iii) Two hyperspheres Si, S2 with centers ci, C2 and radii ri, r*2 intersect orthogonally if and only if (9.23)

|ci - c 2|2 = r\ + r\, which is equivalent to ( 5 1 , 52) = 0. Similarly, a hypersphere S intersects orthogonally with a hyperplane P if and only if its center lies in P :

(9.24)

( c , v ) - d = 0,

which is equivalent to (s ,p ) = 0. (iv) A point x can be considered as a limiting case of a hypersphere with radius r = 0. An incidence relation x G S with a hypersphere S (resp. x G P with a hyperplane P ) can be interpreted as a particular case of an orthogonal intersection of a sphere of radius r = 0 with S (resp. with P ) . We have: x

G

S

<=>

(x, s) = 0,

x

G

P

«=>

(x,p) = 0.

Switching from the Euclidean space R ^ to the sphere corresponds to a different choice of representatives for the points of P (R Ar+1,1): • P o in t y e S N : (9.25)

y z~ ysph = y +

e/v+2-

• Hypersphere S = {y € § n : (*.y) = !}■• (9.26)

s -=

• Great hypersphere S = (9.27)

^Sph

{y

= s + &N+2€z S N : (s ,y ) = 0}

s — ^Sph = :

S+

0 • V N + 2-

Features of this choice of representatives:

9. Foundations

346

(i) In formulas (9.25), (9.26), y and s are points of R N+l with (y, y) = 1 and (s,s) > 1, which is equivalent to y G and s G R ^ 1,1, respectively. Also elements (9.27) (still defined up to a real factor) belong to s G R ^ 1,1. (ii) Incidence relation: y GS

<^>

(y, s) = 0.

Indeed, the relation (y, s) = 0 for y from (9.25) and for s from (9.26) is equivalent to (s,y) = 1. Similarly, the relation (y,s) — 0 for elements s with vanishing ejv+2-component, as in (9.27), is equivalent to (s, y) — 0, which characterizes great hyperspheres. To sum up: in the Minkowski space R ^ 1,1 of homogeneous coordinates, points and hyperspheres (different from hyperplanes) of the Euclidean space R n find their place in the affine hyperplane (£,eoo) = —?>; in particular, R N ~ Q £ = {£ G L n + u :

(9.28) 7ro :

£N + 2

- Ov+i = 1},

R n 3 x h-> x =xeuc = x + eo 4* |x|2eoo —x +

^(| x | 2 — 1) e^ v + i +

^ ( |^|2 +

1) eyv-j-2

GQ

q

(Euclidean metric d£2 H------b d ^ being induced from the ambient R^"1"1,1). The model Q q of the Euclidean space R^ can be viewed as a parab­ oloid in an (N + l)-dimensional affine subspace through eo spanned by e i,..., eyv, ^oo‘ Similarly, points and hyperspheres of (different from great hvperspheres) find their place in the affine hyperplane (£,eyv+ 2) = —1 of the Minkowski space R ^ +1,1; in particular, SN

(9.29)

7ri :

= { t e h N + l '1 : £;v+2 = l},

B y >—> y = ysph = V + e/v+2 €

.

The model of the 7V-sphere can be viewed as a copy of in the ( N + l)-dimensional affine subspace through e#+2 spanned by e j , ..., e^v+iNote that the correspondence between and along the straight line generators of L ^ 1,1 induces the stereographic projection a (compare (9.28) with (9.29) and with (9.14)). In particular, the generators of through the points eo and e ^ correspond to the zero and the point at infinity in R^, and to the south pole yo — —&N+1 and the north pole yoo = e^v+i on S^, respectively. Turning to projective models of lower-dimensional spheres, recall that a hypersphere S in R ^ (or in SN ) can also be interpreted as the set of points x G S, and therefore it admits, along with the representation s, the dual

9.3. Mobius geometry

347

Figure 9.5. Projective model of M obius geometry.

representation as a transversal intersection of P (L Ar+1,1) with the projective TV-space P(5± ), polar to the point 5 with respect to P(L^"+1,1); here, of course, s1- = {x £ : (s ,x ) = 0}. This can be generalized to model lower-dimensional spheres. • Spheres. A fe-sphere is a (generic) intersection of N — fe hyper­ spheres Si represented by s* £ M ^ 1,1 (i = 1 ,..., N — k). Such an intersection is generic if the (TV —k )-dimensional linear subspace of jjiV+1,1 Spanneci by [s space-like: S = sp a n (si,...,s^ _fc) C l ^ 1,1. As a set of points, this /.-sphere is represented as P (L ^ +1,1 fl E-1), where N -k

S1 =

f ] s i = { x e R n+1’1 : ( §i , x ) = ■■■ = ( §N-k, x) = o | i= l

is a (fe + 2)-dimensional linear subspace of R ^ 1,1 of signature (fe + 1,1). Through any fe+ 2 points x \,..., x k+2 G in general position one can draw a unique fe-sphere. It corresponds to the (fe + 2)dimensional linear subspace S 1 = span(xi,...,® fc+2), of signature (fe + 1,1), with fe + 2 linearly independent isotropic vectors x i , ... ,£^+2 £ L ^ +1,1. In the polar formulation, this fesphere corresponds to the (TV — fe)-dimensional space-like linear

9. Foundations

348

subspace k+2 S = P | xj- = 11 G R 7V+1,1 : ( s, x i ) = ■■■ = (s, x fc+2) = o | . 2— 1

To conclude, we mention that for hyperspheres s yet another choice of representatives in is sometimes used: one fixes the Lorentz norm of s. For any k > 0, introduce the quadric (9.30)

L ^ 1'1 = {£ € R n + 1 ,1 : (£ ,0 = k2},

and choose the representative of a hypersphere in L ^ +1,1:

(9.31)

s = s Mob = - ( s + eAr+2) = - (c + e0 + (|c|2 - r 2 ) e 0Q) G L ^ + u . p r

Actually, equation (9.31) contains two representatives of any hypersphere, corresponding to opposite values of p, resp. r, and therefore it represents oriented hyperspheres, each choice of the sign corresponding to one of the two possible orientations of a given hypersphere. Strictly speaking, this choice leads us outside of the projective model of Mobius geometry, and is a remainder of the Lie-geometric approach. We call p G M (resp. r G R ) the oriented spherical (resp. Euclidean) radius of the hypersphere. For any two (oriented) hyperspheres S i, S2, the scalar product of their representatives s Mob is a Mobius invariant: if k = 1, then (5l’ " 2) = ^ ((S1,S2) “ ^ = 2 ^ (r? + ^ " ,C1 " C2'2 is the cosine of the intersection angle of S i, S2, if they intersect, and the inversive distance between S i, S2, otherwise. 9.3.3. Mobius transformations. Mobius geometry is the study of prop­ erties of (nonoriented) hyperspheres invariant with respect to projective transformations of P(MAr“h1’1) which map points to points, i.e., which leave P (L iVH"1,1) invariant. Such transformations are called Mobius transforma­ tions.

Theorem 9.5. (Fundamental theorem of Mobius geometry) a) The group of Mobius transformations is isomorphic to 0 ( N + 1,1)/ {± 7 } ~ 0 + (N + 1,1), the group of Lorentz transformations of M ^ 1,1 pre­ serving the time-like direction. b) Every conformal diffeomorphism of §>N ~ U {00} is induced by the restriction to P (L ^ +1,1) of a Mobius transformation.

The group 0 + (N + 1,1) is generated by reflections, (9.32)

:R JV+U- » R JV+U,

A s{x) = x -

(5, s )

s.

9.3. Mobius geometry

349

These reflections preserve the light cone LA"1 "1,1 and map straight line genera­ tors to straight line generators. Therefore, they induce some transformations on P (L j/v+1,1) ~ Q^, resp. on Q^. The induced transformations on ~ R^ are obtained from (9.32) by direct computations with representatives (9.16) for points and representa­ tives (9.18) for hyperspheres, and are given by (9.13) (inversion in the hy­ persphere S = {x G R ^ : \x —c \2 = r2}); similarly, if s = p is the hyperplane (9.19), then the transformation induced on R ^ by Ap is easily computed to be as in (9.12) (reflection in the hyperplane P = {x G R^ : (u , x) — d}). Similarly, the induced transformations on Q ^ ~ §>N are obtained by a straightforward computation with representatives (9.25) for points and (9.26) for hyperspheres: M{y)

=

( y - s + ^y ^

and so the induced transformation on

s) + ^ ^

ejv+2,

is given by (9.15).

Since (non)vanishing of the eo-component of a point in P(R iV+ljl) is not invariant under a general Mobius transformation, there is no distinction in Mobius geometry between hyperspheres and hyperplanes. The elements of the isotropy subgroup 0+(7V + 1,1) of Lorentz transformations which fix are generated by reflections in the hyperspheres (9.19), which induce reflections in the hyperplanes of R^. Therefore, 0 + (TV + 1,1) is identified with E ( N ) , the group of Euclidean motions of R^. It is convenient to work with spinor representations of these groups. Re­ call that the Clifford algebra G £ (N + l, 1) is an algebra over R with generators e i , . . . , ew +2 G R^"1 "1,1 subject to the relation t v + v t = -2(£, 7?>1 = —2(£, r?),

V£, rj € R N+1,1.

This implies that £2 = —(£,£); therefore any vector £ G R ^ 1,1 \ L ^ +1,1 has an inverse £-1 = —£/(£,£)• The multiplicative group generated by the invertible vectors is called the Clifford group. We need its subgroup generated by the unit space-like vectors: S = Pin+ (iV + 1,1) = {'0 = £i ••-

: <£*2 = - l } ,

and its subgroup generated by the vectors orthogonal to e^: Soo = Pin+(TV+ 1,1) = {?/> = £i •••£„:

= - 1 , (&,eoo) = 0}.

These groups act on R ^ -1"1,1 by twisted conjugations: A^(rj) = In particular, for a vector £ with £2 = —1 one has: M v ) = - C 1^ = M

= rj - 2(£, r?)£,

which is the reflection in the hyperplane orthogonal to £. Thus, 9 is gener­ ated by reflections, while Soo is generated by reflections which fix eo, and

9. Foundations

350

therefore leave Qgf invariant. Actually, S is a double cover of 0 + (7V+ l, 1) ~ Mob(AT), while Soo is a double cover of 0 + (iV + 1,1) — E ( N ) . Orientation preserving transformations from S, Soo form the subgroups K = Spin+(AT + 1,1),

Woo == Spin+ ( N + 1,1),

which are singled out by the condition that the number n of vectors ^ in the multiplicative representation of their elements = £i •• • is even. The Lie algebras of the Lie groups !K and "Kqq consist of bivectors: f)

=

spin(iV + 1, 1) =

fjoo

=

spin00(iV + 1 , 1 )

sp a n je ie ,- : i, j G { 0 , 1, . . . , N , oo}, i ^ j | , =

s p a n je je j : i , j € { 1 , . . . , iV, oo},

9.4. Laguerre geometry Laguerre geometry is a subgeometry of Lie geometry, with hyperplanes dis­ tinguished among all hyperspheres, as the hyperspheres through oo. Thus, Laguerre geometry studies properties of hyperspheres invariant under the subgroup of Lie sphere transformations which preserve the set of hyper­ planes. The following objects in R N are elements of Laguerre geometry. • (Oriented) hyperspheres S = {x G R N : \x — c|2 = r 2} w ith centers c G R n and signed radii r G

M,

can be put into correspondence

w ith ( N + l)-tu p le s (c, r).

• Points x G R n are considered as hyperspheres of radius zero, and are put into correspondence with (N + l)-tuples (x, 0). • (Oriented) hyperplanes P = {x G R N : ( v, x) — d}, with unit normals v G SN_1 and d G M, can be put into correspondence with ( N + l)-tuples (v, d). In the projective model of Lie geometry, hyperplanes are distinguished as elements of P (L j/v"1 "1,2) with vanishing eo-component. Thus, Laguerre geometry studies the subgroup of Lie sphere transformations preserving the subset of P (L iV+1,2) with vanishing eo-component. There seems to exist no model of Laguerre geometry where hyperspheres and hyperplanes would be modelled as points of one and the same space. Depending on which of the two types of elements is modelled by points, one comes to the Blaschke cylinder model or to the cyclographic model of Laguerre geometry. We will use the first model, which has the advantage of a simpler description of the distinguished objects of Laguerre geometry, which are hyperplanes. The main advantage of the second model is a simpler description of the group of Laguerre transformations.

9.4. Laguerre geometry

351

The scene of both models consists of two (N + l)-dimensional projective spaces with dual spaces of homogeneous coordinates, R ^’1,1 and (R ^’1,1)*, which arise from R ^ 1,2 by “forgetting” the eo-, resp. e^-components. Thus, R ^ ’1,1 is spanned by N + 2 linearly independent vectors e i , ... , e^v, ejv+3, ^oo, and is equipped with a degenerate bilinear form of signature (N , 1,1) in which the above vectors are pairwise orthogonal, the first N being space-like: (e?;,ej) = 1 for 1 < i < N , while the last two being time-like and isotropic, respectively: (eN+ 3, eyv+3) = —1 and ( e ^ e ^ ) = 0 . Similarly, (R ^ 1,1)* is assumed to have an orthogonal basis consisting of e i , . . . , eyv, cat+3, ^o, again with an isotropic last vector: (eo,eo) — 0. Note that one and the same symbol (•, •) is used to denote two degenerate bilinear forms in our two spaces. We will overload this symbol even more and use it also for the (nondegenerate) pairing between these two spaces, which is established by setting (eo,eoo) = — in addition to the above relations. (Note that a degenerate bilinear form cannot be used to identify a vector space with its dual.) In both models mentioned above, • Hyperplane P = (v, d) is modelled as a point in the projective space P (R Ar’1,1) with a representative (9.33)

p — v + 2 d e 00 + e^v+3-

• Hypersphere S = (c, r ) is modelled as a point in the projective space P ^ R ^ 1,1)*) with a representative (9.34)

s = c + e0 + re N+3.

Each of the models appears if we consider one of the spaces as a preferred (fundamental) space, and interpret the points of the second space as hyper­ planes in the preferred space. In the Blaschke cylinder model the preferred space is the space P(R 7V’1,1) whose points model hyperplanes P C R^. A hypersphere S C R^ is then modelled as a hyperplane 6 P(R 7V,1,i) : (s,£) = 0} in the space P(R iY’1,1). Basic features of this model are the following: (i) Oriented hyperplanes P C R^ are in a one-to-one correspondence with the points p of the quadric P (L N’1,1), where (9.35)

L ^ ’1,1 = {£ G R ^ ’1,1 : (<£,<£) = 0 }.

(ii) Two oriented hyperplanes Pi, P 2 C R^ are in oriented contact (par­ allel) if and only if their representatives pi, p 2 differ by a vector parallel to e^, that is, if ( pi , p2) = 0 . (iii) An oriented hypersphere S C R^ is in oriented contact with an oriented hyperplane P C R N if and only if if (p, s) = 0. Thus,

9. Foundations

352

a hypersphere S is interpreted as the set of all its tangent hyper­ planes. The quadric P (L iY’1,1) is diffeomorphic to the Blaschke cylinder (9.36)

Z = { ( v , d ) < = R N + 1 : |v| = 1} =

x R c R ^ 1.

Two points of this cylinder represent parallel hyperplanes if they lie on one straight line generator of Z parallel to its axis. In the ambient space R7^ 1 of the Blaschke cylinder, oriented hyperspheres S C R ^ are in a one-to-one correspondence with the hyperplanes nonparallel to the axis of Z: (9.37)

S ~ {(v , d) G R ^ +1 : (c, v) — d — r = 0}.

An intersection of such a hyperplane with Z consists of points in Z which represent tangent hyperplanes to S C R^, as follows from (9.11). In the cyclographic model, the preferred space is the space of hyper­ spheres (R ^’1,1)*, so hyperspheres S C M.N are modelled as points s G P ((R Ar’1,1)*), while hyperplanes P C R N are modelled as hyperplanes {£ : (P>£) — 0} C P((M j/V’1,1)*). Thus, a hyperplane P is interpreted as the set of hyperspheres S which are in oriented contact with P . Basic features of this model are the following: (i) The set of oriented hyperspheres S C spondence with the points (9.38)

RN

is in a one-to-one corre­

c = (c, r)

of the Minkowski space R7^ 1 spanned by the vectors e i , ... , eyv, ejv+3. This space has interpretation as an affine part of P ((R j/V’1,1)*). (ii) Oriented hyperplanes P C R N can be modelled as hyperplanes in R ^ ’1: (9.39)

7r= {(c ,r ) g R ^ ’1 : ((v, 1), (c, r)} = (v, c) - r — d}. Thus, oriented hyperplanes P G R ^ are in a one-to-one correspon­ dence with the hyper planes ir C R ^ ’1 which make angle 7r/4 with the subspace R N = {(x ,0 )} C R ^’1.

(iii) An oriented hypersphere S C R^ is in oriented contact with an oriented hyperplane P C R^ if and only if a G 7r. (iv) Two oriented hyperspheres S i, S 2 C R ^ are in oriented contact if and only if their representatives in the Minkowski space o \, o 2 G R ^ ’1 differ by an isotropic vector: \a\ — a2\= 0. In the cyclographic model, the group of Laguerre transformations admits a beautiful description:

9.5. Pliicker line geometry

353

Theorem 9.6. (Fundamental theorem of Laguerre geometry) The group of Laguerre transformations is isomorphic to the group of affine trans­ formations of R ^ ’1; y i—►AAy + b with A G 0 ( N , 1), A > 0, and b G R ^’1.

9.5. Pliicker line geom etry In this section we denote the homogeneous coordinates of a point x G RP3 by x = (x°, x 1, x2, x3) G R4. For the sake of notational convenience, we abbreviate V = R4. In the standard way, projective subspaces of RP3 are projectivizations of vector subspaces of V. In particular, let x,y G RP3 be any two different points, and let x, y G V be their arbitrary representatives in the space of homogeneous coordinates. Then the line g = (xy) C RP3 is the projectivization of the two-dimensional vector subspace span(x, y) C V. After H. Grassmann and J. Pliicker, the latter subspace can be identified with (a projectivization of) the decomposable bivector (9.40)

g —x Ay

G

A 2 V.

We choose a basis of A2V to consist of e* A ej with 0 < z < j < 3. A coordinate representation of the bivector (9.40) in this basis is (9.41)

g = ^ 2 g lJ^i A ej,

gtJ = x ly3 - x 3 y\

(ij)

The numbers (g 01 ,g 02, g03, g12, g 13, g23) are called Pliicker coordinates of the line g. They are defined projectively (up to a common factor). Indeed, changing the choice of the two points defining g from x, y to x, y with the homogeneous coordinates x-7 = ax-7 + byi, y*7 = cx*7 + dj/-7, ad — be ^ 0, would lead to a simultaneous multiplication of all gli by a common factor: gli — (ad — bc)gli .

Not every bivector represents a line in RP3, since not every bivector is decomposable, as in (9.40). An obvious necessary condition for a non-zero g G A2V to be decomposable is (9.42)

£ A
It can be shown that this condition is also sufficient. In Pliicker coordinates, this condition can be written as (9.43)

9° y 3 - < / v 3+ / y 2 = o.

Summarizing, we have the following description of £ 3, the set of lines in RP3, within Pliicker line geometry. The six-dimensional vector space A2V

354

9. Foundations

with the basis e j Ae ^ is supplied with a nondegenerate scalar product defined by the following list of nonvanishing scalar products of the basis vectors: (e0 A ei, e 2 A e 3) = — (e0 A e 2, e L A e 3) = (e0 A e 3, e i A e 2) = 1.

It is not difficult to verify that the signature of this scalar product is (3, 3), so that we can write A2V ~ M3,3. Denote (9.44)

L3’3 = { g e A 2F : ( M ) = 0}.

The points of the Pliicker quadric P (L 3,3) are in a one-to-one correspondence with elements of £ 3. A fundamental feature of this model is the following: • Two lines g, h in MP3 intersect if and only if their representatives in A2V are polar to one another: (9.45)

{g, h) = g01 h23 - g 02h 13 + g 03h V2 + g 23 h01 - g l 3 h02 + g 12 h03 = 0.

In this case the line £ C P(A 2]/) through [g] and [h] is isotropic: P (L 3,3).

£C

To prove this, note that if the lines g , h intersect at the point 2 , then g — x A z and h — y A z, and then g A h. = 0. Conversely, if the lines g, h do not intersect, then their lifts to V span the whole of V , and so g A h, ■£ 0. It remains to observe that g A h = (g, h) eo A ei A e 2 A e.3. Next, we turn to important linear subsets of the Pliicker quadric. • Any isotropic line f C P(JL3,3) corresponds to a one-parameter fam­ ily of lines in MP3 through a common point, which lie in one plane. Such a family of lines is naturally interpreted as a contact element (a point and a plane through this point) within the line geometry. • Other than in Lie geometry, in the present case of signature (3,3) there exist also isotropic planes, which are projectivizations of 3­ dimensional vector subspaces of K2V that belong to L 3,3. There are two sorts of isotropic planes in the Pliicker quadric P (L 3,3). A 11 isotropic plane can represent: a) a two-parameter family of all lines in MP3 through some com­ mon point; such a family is naturally identified with that com­ mon point; (5) a two-parameter family of all lines in some plane in MP3; such a family is naturally identified with that common plane. To see why the latter statement holds, consider three noncollinear points in the isotropic plane. Their pairwise connecting lines are all isotropic. Therefore these three points represent three pairwise intersecting lines in MP3. If all three are concurrent, then we are in the situation a). Otherwise they lie in a plane in MP3, and we are in the situation (3).

9.5. Pliicker line geometry

355

Projective transformations of P(R 3,3) which leave the Pliicker quadric P (L 3,3) invariant can be distinguished depending on their action on the two types of isotropic planes. Theorem 9.7. (Fundamental theorem of Pliicker line geometry) a) The group of projective transformations of RP3 is isomorphic to the subgroup of 0 (3 ,3)/(±/) consisting of transformations which preserve the types a ) and (3) of the three-dimensional vector subspaces in L 3,3. b) The group of correlative transformations of MP3 is isomorphic to the subgroup of 0 (3 ,3)/(±J) consisting of transformations which interchange the types a ) and (3) of the three-dimensional vector subspaces in L 3,3. Next, we discuss planar families of lines. Such a family of lines is repre­ sented by a conic section P (E n L 3’3), where E stands for a three-dimensional vector subspace of R3,3 such that the restriction (•, -)|s is nondegenerate. It is not difficult to realize that four pairwise nonintersecting (skew) lines in RP3 belong to a planar family (have linearly dependent representatives in L 3,3) if and only if they belong to a regulus (one family of generators of a ruled quadric in RP3, i.e., of a one-sheet hyperboloid or of a hyperbolic paraboloid). The complementary regulus is represented by the dual planar family of lines P(S~L n L 3’3). Finally, we briefly mention the duality in Pliicker line geometry. One can describe any projective subspace P(E) C RP3 through its dual subspace P(E-l ) C (RP3)*, where E1- C V * is the annihilator of the vector subspace E C V. As a set of points, P(E) is the intersection of planes represented by P (E ± ). Thus, a plane u C RP3 can be described through an element of P (E X) C (RP3)* with homogeneous coordinates u = (uq , u i , ^2,^3) = E f= o ^ e? ^ ^** As a se^ ° f points, this plane consists of homogeneous coordinates x = (x°, x 1, x 2, x 3) £ V satisfying

x

G RP3 with

3

(9.46)

^

U[Xl — 0.

1=0

This description of u is dual to the description as the projectiivization of the three-dimensional vector subspace span(x, y,£) C V, where x , y , z are homogeneous coordinates of any three noncollinear points x , y , z € u. In the spirit of the Grassmann-Plucker approach, the latter vector subspace can be represented by (a projectivization of) the decomposable three-vector u = x A y A z E A 3 V . In the basis of A3V consisting of e^ A ej A e&, 0 < i < j < k < 3 , one has: u = ^ (■ijk )

u% i k ei A e j A e

,

u^k =

Xi

Xj

xk

Vi

Vj

Vk

Zt

Zj

zk

356

9. Foundations

It is easy to see that the homogeneous coordinates ui can be normalized so that uo = u 123, 'Ui = —u023, U2 = u 013, us — —u012. Similarly, in the dual description, any line g C MP3 can be viewed as an intersection of two planes u,v C MP3, and thus can be described through span( u,v) G V*, which, in turn, can be represented by (a projectivization of) the bivector g = u A v e A 2 ( V*) .

(9.47) In coordinates: (9.48)

g = uAv = y^gij‘

AeJ,

g ij — U t V j

U jV i

(ij)

The sextuple of numbers ( 501,502>503»512, 513, 523) is called dual Pliicker coordinates of the line g. Remarkably, this new set of coordinates is related to the previously introduced Pliicker coordinates in a fairly simple way: if u,v are any two planes in RP3 intersecting along the line 5 , then their homogeneous coordinates Ui, Vi can be so normalized that the dual Pliicker coordinates (9.48) of the line g = u fl v coincide, after a suitable reordering, with its coordinates (9.41): (9.49)

713

5oi = 523,

502

523 =

513 = - 5"

.01

/ 5 .02

12.

= 5 012 = 503503

To see this, take g = (xy) = u D v and choose points p G w, q G v so that p 0 v and q £ u. We can normalize homogeneous coordinates of the planes u, v so that /x° y° p°

1 x 2 x z\ yl y2 y3 p 1 p 2 p3 x

(u 0 Ul

vo\ Vi

U2

n

/0

0\ 0 0 0 1

V3 J 0/ \9° 91 Q2 Q3) V Now (9.49) follows from a well-known generalization of the Cramer rule, which says that the 2 x 2 determinants gli = x lyi —xhy1 are proportional to the 4 x 4 determinants obtained from the matrix of the latter linear system by replacing the i -th and j -th columns by the columns on the right-hand side of the system. \U3

To conclude, we mention a couple of useful relations for the usual and dual Pliicker coordinates of lines. They follow directly from definitions. • Homogeneous coordinates of the plane u through a line g and a point p g are given by 3

(9.50)

Uj = ^ g jkpk k=0

(j = 0,1,2,3);

9.6. Incidence theorems

357

homogeneous coordinates of the intersection point x of a line g with a plane v not containing g are given by 3

(9.51) • A point x belongs to a line g if and only if 3

(9.52) k= 0

a line g lies in a plane u if and only if 3

(9.53) By the way, the last statement allows us to give a simple argument for the claim that any g £ L 3,3 corresponds to a line in MP3. Indeed, (g,g) = 0 and g 0 is equivalent to the fact that the rank of system (9.53) is equal to 2, and so the solution of this system delivers two different planes. They intersect along the line we are looking for. 9.6. Incidence theorems This section contains a collection of classical incidence theorems which lie in the basis of discrete differential geometry. We will often use the cross-ratio of four collinear points a, 6, c, d, defined as (9.54)

q{a, 6, c, d) =

/(a, b)

l(c,d)

/(6, c)

l(d,a)

and the fact that the cross-ratio is invariant under projective transforma­ tions. 9.6.1. Menelaus’ and Ceva’s theorems. Theorem 9.8. (Menelaus’ theorem) Consider a triangle A ( A \ A 2 As) in the plane. Let P\2 , P 23 , P 31 be some points on the side lines ( A 1 A 2 ), ( A 2 A 3 ), ( A 3 A 1 ), respectively, different from the vertices Ai of the triangle. These three points are collinear if and only if

(9.55)

l(Au Pi2) 1{A2,P23) l(A3,P3l) l(Pi 2 ,A 2 ) ' K P 23 , As)'l(Pbu Ax)

Theorem 9.9. (Ceva’s theorem) Consider a triangle A ^ i ^ A a ) in the plane. Let P \ 2 , P 2 3 , P 3 1 be some points on the side lines (A 1A 2), (^ 2^ 3); ( A 3 A 1 ), respectively, different from the vertices Ai of the triangle. The three

358

9. Foundations

F i g u r e 9.6. M enelaus’ theorem.

lines (A 1 P 23 ), (A 2 P 31 ), {A 3 P 12 ) have a common intersection point if and only if (

,

l ( A2, P 23) . l( A3,P3l) = 1{P\2,A2) 1{P2Z,A3) 1{P31,Ai )

1(Ai , P i 2)

F i g u r e 9.7. Ceva’s theorem.

Both Menelaus’ and Ceva’s theorems have a similar flavor: their hy­ potheses are of a seemingly affine-geometric nature (the left-hand sides of equations (9.55), (9.56) are expressed in terms of quotients of directed lengths), while their conclusions are projectively invariant. Actually, one can show that the numeric value of the cyclic product of the quotients of directed lengths on the left-hand sides of (9.55), (9.56) is itself a projectively invariant quantity. This is a consequence of the following theorem. Theorem 9.10. (Projective invariance of a cyclic product of di­ rected lengths ratios for a triangle) Consider a triangle A ( A i A 2 As) in

t^ L h s id e n c e jh e ^ x

the plane, and let P l9 P (A

m

i

A ' }d

n ~

y

p

,

-

a) Z?eno<e 6yQ,2 the ;«/Pr,e ,• ' " F u jm ” « •» ............« “ » / » » » <

*

*

•

(A ,A 2 )-

* ° f the triangle. tee M M 2)

(9.57)

b) 5 ,< o

lu t

i(P 2 3 " 4 3 )

“ c

°,t t K ,a e

(9-58)

/TB1^ 12'

•^ 2)

1^ 3) _l(Aa, P n )

l (p 2 3 , A 3)

K ^ A i ) = q (A u P v 2 , A 2 , R 12).

< **> .

360

9. Foundations

Proof. Clearly, this theorem yields Menelaus’ and Ceva’s: the cross-ratios on the right-hand sides of equations (9.57), (9.58) are equal to 1 if and only if Q 12 = ^ 12, resp. R \2 = P i 2- For a proof, note that, since both sides of (9.57), (9.58) are invariant under affine transformations, it is enough to consider A\ = (0,0), A 2 = (1,0), As = (0,1), and then P 12 = (xi,0), P 23 = (1 - x 2 , x 2), P 31 = (0,1 - xs) with some x\, x 2 , x 3 G R \ {0,1}. Then a straightforward computation confirms both claims of the theorem. □ 9.6.2. Generalized Menelaus’ theorem. Upon using the results of The­ orem 9.10 to “cut off vertices” , one can prove the following result. Theorem 9.11. (Projective invariance of cyclic products of directed lengths ratios) Let A\, A 2, . . . , A n be n > 3 points in such that no three consecutive points in cyclic order are collinear. Let P 12, P 23, • • •, Pni be points on the lines ( A i A 2), ( A 2 As ) , ..., ( AnA\), respectively. Then the product of ratios of directed lengths

TT

Pj,i+ 1)

is invariant under projective transformations.

The geometric meaning of the situation where the cyclic product in Theorem 9.11 takes a special value ( —l ) n is given by the following result. Theorem 9.12. (Generalized Menelaus’ theorem) L e t A \ , . . . , A n be n points in general position in Rn-1; so the affine space through the points Ai is (n — 1) -dimensional. Let P ij+ i be some points on the lines ( Ai A{ +i ) different from A {, Ai+\ (indices are taken modulo n ). The n points P ij+ i lie in an (n —2) -dimensional affine subspace if and only if the following relation fo r the quotients of the directed lengths holds: TT

1)

_

AAz(PM+i,A i+1)

k

1 '

Proof. The points P^i+i lie in an (n — 2)-dimensional affine subspace if there is a nontrivial linear dependence n

E

n

M P i,i+1 = 0

with

i= l

^

IM = Q-

i= 1

Substituting Pi,i+i = (1 — £i)Ai + & A i+ 1, and taking into account the gen­ eral position condition, which can be read as linear independence of the vectors A iA i , we come to a homogeneous system of n linear equations for n coefficients fii\ tiHi + ( 1 - £i+ i)/Xt+i = 0,

i = 1,..., n

361

9.6. Incidence theorems

(where indices are understood modulo n). solution if and only if

Clearly it admits a nontrivial

= (- i)”

□

(Menelaus’ theorem corresponds to n = 3.) 9.6,3. Desargues’ theorem.

Theorem 9.13. (Desargues’ theorem) Two triangles A ( A \ B i C i ) and A ( A 2 B 2 C 2) in a plane are perspective from a point if and only if they are perspective from a line. This means that the three lines (A 1 A 2 ), (B 1 B 2 ), (C 1 C 2 ) are concurrent if and only if the three intersection points ( A\Bi ) fl ( A 2 B 2 ), {A\C\) fl ( A 2 C 2 ), and { B 1 C 1 ) ^ { B 2 C 2 ) are collinear.

Proof. For an illustration, see Figure 9.10. To find a criterion for our tri­ angles to be perspective from a point, we denote Q = ( A \ B i ) n ( A 2 B 2), R = (.B\C\) fl (B 2 C 2 ) and consider the two triangles A ( B \ Q B 2 ) and A ( B i R B 2), intersected by the lines (A\A2) and (C\C2), respectively. Menelaus’ theorem yields a necessary and sufficient condition for the lines ( A i A 2) and (C iC 2) to intersect ( B 1 B 2 ) at one and the same point O: l ( B u A x)

l ( Q , A 2)

/(Si, O)

/(Si, C i)

l ( R , C 2)

l ( A u Q)

l ( A 2 , B 2)

1(0, B 2)

l ( C u R)

Z(C2,S 2)

or (9.59)

Z(Bi,j1i)

l ( Q , A 2)

1{B2 ,C 2)

l ( R , C 1)

l ( A u Q) ' l ( A 2 , B 2) ' l ( C 2 , R ) “ /(Ci,BO

- 1.

9. Foundations

362

Similarly, to find a criterion for our triangles to be perspective from a line, we consider the triangles A ( Q B \ R ) and A ( Q B 2 R) intersected by the lines (.A\C \) and ( A 2 C 2 ), respectively. Menelaus’ theorem yields a necessary and sufficient condition for the lines ( A\C\) and ( A 2 C 2 ) to intersect ( QR ) at one and the same point P : l(Q,Ai)

l(Bu Ci) _

l ( A x, B{ ) ’ l ( C u R)

l ( P , Q ) __ l ( Q , A 2) l(R, P )

l ( B 2, C 2)

l ( A 2 , B 2) ‘ l ( C 2 , R ) ’

or rofim 1' ’

l { Q' A l ) l ( Ai , B\)

l ( C\, R)

l(R ' c ^ 1{C2 , B 2)

i ( B2, a

2) _ l(A2, Q )

•

By inspection, equations (9.59) and (9.60) are identical.

□

A good way to understand Desargues’ theorem is to regard Figure 9.10 as a plane projection of a truncated three-dimensional pyramid with the tip O. Then it becomes clear that the points P, Q, R belong to the intersection of the planes ( Ai B\Ci ) and ( A 2 B 2 C 2 ) and are therefore collinear. 9.6.4. Quadrangular sets. Definition 9.14. (Quadrangular set) Let A, J3, C, D be four points in a plane II, such that no three of them are collinear. A complete quadrangle with these four points as vertices consists of six lines connecting six pairs among the points A , B, C, D. The lines in each pair ( A B ) and ( C D ) ,

( AC ) and (B D ),

( A D ) and ( B C )

are called opposite. Let £ be a line in the plane II not containing any of the points A , B , C , D . The three pairs (Pi, Q\\P2, Q 2 ', P3, Q3) of intersection points of i with the three pairs of opposite lines of a quadrangle are said to form a quadrangular set.

One should note the symmetry of this definition with respect to any combination of flips Pi Qi within any pair, as well as with respect to any permutation of the numbers 1, 2, 3. Theorem 9.15. (On quadrangular sets) A quadrangular set of points (Pi, Q i; P2, Q2; P3, Q 3 ) is characterized by the relation

('ofii'i (9.61)

n 2,P3,
l ( p 3i Qi )

Kp 2,Q3)

1

Any five points of a quadrangular set determine the sixth point uniquely.

9.6. Incidence theorems

363

P roof. Let O = ( A C ) fl ( B D ) (see Figure 9.11 for an illustration and for other notation). Projecting the line ( A C ) to £, first with the point B and then with the point D as a center, we find the following two identities:

q ( A , C , 0 , Q s ) = q( P\, Q 2 ,P?>,Q?),

q ( A , C , 0 , Q 3) = tf(P2, Qi, ^ 3, Q 3).

As a consequence, relation (9.61) holds.

□

One can extract various consequences from this proof. For instance, it follows that the points of a quadrangular set always build three point pairs of a projective involutive self-map of L

We will mainly use the following

corollary:

Theorem 9.16. (Pappus’ theorem on quadrangular sets) Let a quad­ rangular set of points (Pi, Q i; P2, Q2] P3, Q3) on a line £ be given. Take any plane II through this line, and let A, B , C be three points in II such that each of the lines ( AB ) , ( B C ) , ( A C ) passes through one of the points of the first, resp. the second, the third pair of the quadrangular set. Then there exists a unique point D such that the opposite lines ( C D ) , ( A D ) , ( B D ) of the com­ plete quadrangle with vertices A, P , C, D pass through complementary points of the first, resp. the second, the third pair of the quadrangular set. This is illustrated in Figure 9 . 12: if (Pi, Q i; P2, Q2; -P3, Q3) is a quad­ rangular set coming from a complete quadrangle with vertices A , B , C , D, and the points A f, B ' , C f are given such that P\ G ( AfB f), P 2 G ( B fC f) and Q 3 G (A 'C 1), then there is a unique point D f such that Q\ G ( C fD f),

Q2

( A ' D f), and P3 G ( B fD f). This means that the lines (QiC"), ( Q 2 A f) and ( P 3 B' ) have a common intersection point D f. G

364

9. Foundations

B

F i g u r e 9.12. Papp u s’ theorem.

9.6.5. C arnot’s and Pascal’s theorems. The following two theorems give two different (but, of course, equivalent) characterizations of six points lying on a conic. Theorem 9.17. (C a rn ot’s theorem ) Let a pair of points { P i j , Q i j ) be chosen on each side line ( A { A j ) of a triangle A ( A i A 2 As). The six points Fiji Qi j lie on a nondegenerate conic if and only if the following relation

holds:

(9.62) l {Ai ,Pi 2)

l( A2, P23 )

ljA- 3 , P 3 1 )

l ( A i , Q i 2)

1(^2, Q 23)

I { A 3 , Q 31) _ ..

l{ Pi 2 , A 2)

1(P23,As)

l(Psi,Ai)

l{Qi2iA 2)

1{ Q2z,Az)

1 { Q ^ A X)

see Figure 9.13.

F i g u r e 9.13. C arnot’s theorem.

’

9.6. Incidence theorems

365

P roof. This claim is almost trivial for the case when the conic is a circle. Indeed, in this case the product P i j ) * l( Ai, Q i j ) =

Pik) ’ 1{A{, Qik)

is nothing but the degree of Ai with respect to the circle. Theorem 9.11 assures that condition (9.62) is projectively invariant. Therefore the claim of the theorem holds for all conics that are images of a circle under projective transformations, that is, for all nondegenerate conics. □ Theorem 9.18. (Pascal’s theorem ) A planar hexagon with vertices P i , ... , P q is inscribed into a conic if and only if the intersection points of the opposite sides of the hexagon, Bi = ( P M

b 2 = ( P 2 P 3 ) n (p5p6),

n (p4p5),

b 3 = (p 3 p 4) n (p6f t ),

are collinear.

P roof. Introduce three auxiliary points A 2 = (P 3P 4) n (P XP2),

A i = (P 1P 2) n (P 5P6),

A 3 - (P5P6) n (P3P4);

see Figure 9.14. Applying Menelaus’ theorem to the triangle A ( A \ A 2 A 3)

Figure 9.14. Pascal’s theorem.

intersected by the lines (P4 P5 ), (PePi), and (P2P3), we find the following relations: KAuBr)

l ( A 2 ,P 4)

l(A 3 , P 5)

l ( B i , A 2)

l ( P 4 , A 3) ' l{P*,,Ax)

l(A 2 , B 2)

l ( A 3, P 6)

IjAuPj)

=

’ =

l ( B 2 , A 3) ' l ( P 6 , A i ) ' l ( P u A 2) l ( A 3 , B 3)

1(Au P 2)

l ( A 2 ,P 3)

l{B 3 , A i )

l ( P 2 , A 2) ' l ( P 3 , A 3)

’ =

’

366

9. Foundations

We multiply these three equations and take into account that, according to Carnot’s theorem, 1(A2,P a)

l ( A 3, P 5)

1( A u P 2)

1( A 2, Ps)

l ( A 3, P 6)

l(Ai,P\) _

l ( P 4, A 3) ' /(P5, A\) ' l ( P 2, A 2) ' l ( P 3, A 3) ' l ( P 6, A i ) ' l ( P i , A 2)

'

Therefore, we arrive at 1{A\,Bi )

l ( A 2, B 2)

1{A s , B s) _

l ( B u A 2) ' 1( B 2i A s ) ' l ( B s, A , ) ~

‘

But, according to Menelaus’ theorem, the latter relation is equivalent to J3i, B 2, B s being collinear. □ 9.6.6. Brianchon’s theorem. Theorem 9.19. (Brianchon’s theorem) A planar hexagon with vertices F i , ... ,Pq is circumscribed about a conic if and only if its three diagonals (P 1 P 4)} (P 2 Pb)> and (P 3 P 6 ) are concurrent; see Figure 9.15. Proof. This theorem is projectively dual to Pascal’s theorem.

□

An important particular (or, rather, degenerate) case appears when two opposite points Pi lie on the conic. Corollary 9.20. I f a planar quadrilateral is circumscribed about a conic, so the lines (P\P2), (P 2P 3), (P 3P 4), (-P4-P1) are tangent to a conic, with tangency points M\, M 2, M 3, M 4, respectively, then the diagonals {P 1 P 3 ) and (P 2^ 4), as well as the lines connecting the opposite tangency points, (M 1M 3) and (M 2M 4), are concurrent; see Figure 9.16.

9.6. Incidence theorems

367

F i g u r e 9.16. Degenerate case of Brianchon’s theorem.

9.6.7. M iqu el’s theorem. Theorem 9.21. (M iq u el’s theorem ) Consider a triangle with vertices f i , f 2j f$j and choose a point fij on each side ( f i f j ) - Then the three circles nCj k through ( f i , f i j , f i k ) intersect at one point /123.

F i g u r e 9.17. M iquel’s theorem.

P roof. Denote the angles of the triangle A (/i, f 2, fs) by a 1 , a 2, oc3 , respec­ tively. The circles T1 C 23 through (/1, /12, /13 ) and t 2C \z through (/2, /12, / 23) intersect at two points, one of them being f i 2. Denote the second intersec­ tion point by /123. We have to show that this point /123 belongs also to the

368

9. Foundations

circle T3C12 through (/3,/13,/23)- For this, note that A/123/12, /123/13) = 7T — C*l,

^ (/ l 23/ l 2 , /123/23) = 7T - 02,

as it follows from the circularity of the quadrilaterals (/1, /12, /123, /13) and (/ 2 , /12, /123, / 23). A s a consequence, we find: ^(/l23/l3, / 123 /23 )

= 27 T — ( 7T — c * l) — ( 7T — (*2) = C*1 + C*2 = 7T -

03,

and this yields that the quadrilateral (/ 3 ,/13,/123,/23) is also circular. See Figure 9 . 17 . □

Appendix. Solutions of Selected Exercises

A .I. Solutions of exercises to Chapter 2

2 .2 . Denote the intersection points as follows (see Figure A .l):

4 = (xx^ n (xkxik),

4

-

(XjXij) n (xjkx m ).

F i g u r e A . I . Elementary hexahedron of a planar Q-net.

369

Appendix. Solutions of Selected Exercises

370

The condition for A\, A ^ A ^ A 2\ being collinear dictates the following construction of the point X123 from the seven points X , X u X tJ: find the intersection points

A\ = {X X i) n (X 3 X 13),

a %=

( X X 2) n (x 3x 23),

draw the line (A^A^) , find its intersection points with the lines (X2X \ 2) and (X1X12):

a 312 = (X2X 12) n (AfAl),

Alx = (X\X\2) n (a?a 2),

and finally find X123 as the intersection point

x 123 = ( ^ 2x 23)n (A | 1x 13). It remains to prove that one gets the same result if one bases the construction 011 either of the conditions that the points A 2, A^, A 23, A%2 be collinear or the points A \ , A 3, A^x be collinear. The idea of the proof is to construct a three-dimensional figure for which the original figure is a planar projection (the same idea is commonly used to prove Desargues’ theorem). Let a be the plane of our figure, sup­ pose that A\, A 2, A\2, A 2l lie on a common line, and let (3 be some other plane through this line. Choose some point O outside of both planes, and define the quadrilateral (Y3, Y13, ^123, ^23) in ft as the projection of the quadrilateral (X3, X13, X12.3, X23) from O onto ft. All faces of the hexa­ hedron (X , X\, X\2, X 2,V3, Y13, Vi23? V23) are planar. For example, the face (X, X 2, V23, V3) is planar because the lines (X3X23) and (V3V23) meet in A 2 by construction and this is also the intersection point of the lines (X3X23) and ( I I 2). Now, consider the points

A^ = ( x x 2) n ( X ! X l2),

Bis = (^3^23) n (YiaYm),

b\ = ( x y 3) n ( x ^ ) . B\2 = {X 2Y23) n (X l2Yl23).

They belong to the intersection of the planes (XX2V23V3) and (X1X12 V123V13) and therefore are collinear. Hence, the points A 2, A\, A23, A ^ , which are the projections of these points from O onto a, are also collinear. The collinearity of the points A\ , A3, A§j is proven analogously.

2.3. The combinatorial structure of the situation in question is that of a hypercube; cf. Figure A .2. One has to prove that the four planes combi­ natorially corresponding to the four “horizontal” elementary squares in this figure have a common point. One can do this by the same argument as in the proof of Theorem 2.5, first assuming that the whole configuration lies in a four-dimensional space, and then performing a regular limit passage to

A.I. Solutions of exercises to Chapter 2

371

the three-dimensional situation. In the four-dimensional case one can argue as follows: the intersection of four planes n n n (1) n n (2) n n (12) can be alternatively represented as the intersection of four three-spaces y ( °,1) n y ( 0,2) n ^(1,12) n y(2,12)^

where the space with G {0,1, 2,12} is spanned by the planes 11^ and nW of the corresponding quadrilaterals in the nets f ^ and An intersection of four three-spaces in R4 consists generically of one point. ,( 12)

J2

f (12)

J12

Figure A . 2. Quadrilateral (/, /i, / 12, /2) and those corresponding to it in the nets

/ (2\ and / (12\

2.4. From equations (2.10)-(2.12) we derive the following formulas:

(A-1} (A .2)

+Ti(^)){1+^) =1+{1+Cij)^ +(1+Cji)h

(x

(1 + Tibj)( 1 + bi) = 1 + (1 + ctj)bj + (1 + cji)bi-

The symmetry of their right-hand sides implies that equations (2.79) are compatible and therefore determine the functions 0, : Zm —> R (associ­ ated to points of Zm) uniquely up to constant factors, which can be fixed by requiring 0(0) = 0+ (O) = 1. Moreover, (A .l), (A . 2) imply that the functions 0, (j)+ satisfy the equations (A .3)

6iSj(/>

—

Cij5j(f) + Cj

(A.4)

SiSj4>+

=

c J ^ + + 4 ^ +,

372

Appendix. Solutions of Selected Exercises

for all 1 < i 7^ j < rn. The solution 0 of equation (A .3) is directly specified by the initial data (F^), via integrating” the first equation in (2.79) along the coordinate axes 2V Introduce the functions p : Zm —» R N and ^ : Zm —> K by formulas (1.20), so that the classical representation (2.75) will remain valid. A direct computation based on (2.9), ( 2 .10)—(2 .12), and (2.79) shows that formulas (2.76), (2.78) hold with /a _\

O/i 1 a* = — — . n
(A.5)

Finally, check that the quantities a* satisfy (2.77). The solutions a* to this system are uniquely specified by the values on the corresponding coordinate axes which, in turn, can be obtained from the values of + on coordinate axes, yielded by the data (F^) of the fundamental transformation. Now the solutions p, if) of equations (2.76), (2.78) with initial datap(O) = /+ (0) —/(0) and ^ ( 0) = 1 can be found. 2.5. The consistency is expressed as Tifagk) — Tj(rigk) and is assured by the equation (A.6)

(1 +

(1 +

T i C j k ) ( l + Cik) =

Tj Cik) (

1+

C j k) ,

which, in turn, is a consequence of (2.7). Indeed, (2.7) can be equivalently rewritten as (1 + TiCjk)( 1 + Cik) — 1 + (1 + 7~kCij ) Cjk + (1 + TkCji)Cik , with the right-hand side symmetric with respect to the flip i

j.

2.7. One has to prove the compatibility condition for the system (2.80), Ti(rjdk) = Tj(ri6k)- In other words, one has to demonstrate that

is symmetric with respect to i and j . The latter expression is computed straightforwardly with the help of ( 2 .22): j _ 7a + w 1

. 1

Ij klkj

_ 1~

lijlji

(1 _ W t j )

~

Ijlllj

Ifjklfkj

~

)

Ikilih

~

l i j ^j j k' lki

~

Hji'lfik'lkj

1 — Ijilij

and its symmetry is apparent. 2.16. Suppose the sides of one of the quadrilaterals are represented by the complex numbers a, 6, c, d, while the sides of the second are represented by aa, fib, 7 c, 8d, with a, /3,7 ,5 G M. Thus, a + b + c + d — 0,

aa + /3b + 7 c + Sd — 0 .

A.I. Solutions of exercises to Chapter 2

373

The complex cross-ratios of the quadrilaterals are equal if and only if <27 =

fiS. We only have to show that this equality yields the duality of the quadri­ laterals, i.e., the parallelism of the noncorresponding diagonals: a + b || fib + 7c,

b + c || a a + fib.

These two relations are demonstrated in the same way. For instance, for the first we have

( a — 5)a + ( fi — 5)6 + ( 7 — 5) c = 0

(5 — a ) ( a + b) = (fi — a ) b + ( j

— 8) c.

It remains to use the equality (fi — a ) : fi = (7 — 5 ) : 7 , which follows from 0 7 = (35.

2 .20 .

For any quadruple (Pi, P2, P3, P4) of planes in R P 3 with a common

point, one can introduce the notion of the “diagonal plane” as the plane spanned by the lines P\ fl P3 and P 2 fl P4.

Now the projective dual of

Theorem 2.26 can be formulated (and turned into a definition): a Q*-net P

: Z 2 —> {planes in R P 3} is called a dual Koenigs net if the diagonal

planes of the four quadruples (P, Pi, Pij, P j ) with (i, j ) G {(=L1, ± 2)} meet at a common point.

The criterion obtained by the projective dualization

of Theorem 2.27 states that a Q*-net P : Z 2 —> {planes in R P3} is dual Koenigs if and only if for every u G Z 2 the three points

x (Up) = pnPi2nP_i,2,

x (down) = PnPi _2nP_i _2,

= PnPinP_i

are collinear, or, equivalently, if the three points

X 0eft) = pnp_i,2np_i -2, are collinear.

x (right) = PnPi,2nPi _2,

x {2) = PnP2nP_2

There exists yet another equivalent formulation of this cri­

terion. Denote by X ^ \

(i , j ) G {(d b l,± 2 )}, the intersection point of the

four planes ( P, Pi, Pij, Pj ) .

Thus, all four points

lie in the plane P .

Considering the Q*-net P as a Q-net, these point would be the vertices of the elementary quadrilateral lying in the plane P .

The diagonals of this

quadrilateral are the lines

^(+) = ( x (_1’“ 2)x (1’2)),

6 ^ = ( x (_1’2)x (1’“ 2)).

According to Brianchon’s theorem, a Q*-net P : Z 2 —> {planes in R P3} is dual Koenigs if and only if for every u G Z 2 the six lines

6+\ 6~\ and l {ij) = P n

(i,j)€ {(±l,±2)},

are tangents to a common conic in the plane P .

2 .21 . According

to Theorem 2.32, equation for the net / can be taken in

the form f\2 — f = a ( f 2 — f i ) , so that the formulas of Exercise 2.15 yield

374

Appendix. Solutions of Selected Exercises

h = a a - 2 and k — aa-\. As for the net M, formulas (2.36) show that in suitable homogeneous coordinates M, namely M = ( - - - ) ( M , 1), V V\

V2 )

its equation reads A/12 — M = a2M 2 —a\M \, so that h — a\a and k = a2a.

2.22. We can assume that the system of homogeneous coordinates in RP2 is chosen so that / - (0,0,1),

/1 - (1,0,1),

f 2 = (0,1,1),

f l2 = (1,1,1).

Then L i = (1,0,0),

L 2 = (0,1,0),

and we can assume that L 3 = (£3*0 , 1),

U = (£4, 1, 1),

Lb = (0,^5,1),

Lq = (1,^6, !)•

Now it is easy to compute the condition for L i , ..., L q to lie on a conic: it reads

^ 5 ( 6 - 1X& - 1) = Z& iU - l)(& - !)• At the same time, we have: q ( f , L u f u L 3) =

q ( f 2, L 4, f 12, L 1) = - A - ,

?3

^4-1

q ( f , L 2, f 2, L 5) =

q ( f i , L q, /12, L 2) = s5

s6 — 1

This proves the claim. 2.23. The lines £ = (M M + ) can be assigned to elementary hexahedra of F . Each £ is the intersection of the “black” and the “white” planes of the corresponding hexahedron. Moreover, two lines corresponding to two neigh­ boring hexahedra share a point which is the intersection point of diagonals of the common face. Thus, the sextuple of points on £ consists of intersection points of diagonals of all six faces of the corresponding hexahedron (Af, A f+ come from the bottom and the top faces, and the other four points come from the four side faces). It has been shown in the proof of Theorem 2.29 that these six points form a quadrangular set. 2.26. If a T-net is considered as a particular Q-net, then the coefficients Cij of the Laplace equation of the Q-net are given by — I + Cij — —aji. Equation (2.16) for rotation coefficients 7ij yields: _ 7 j7j

_

Cji

^ -f

dj)(

1+

Cj i )

A.I. Solutions of exercises to Chapter 2

375

Now (2.84) yields

A comparison with (2.81) shows that one can take p = a2. 2.27. The system of difference equations (2.90) is compatible due to the first equation in (2.59). A solution 9 : Z2 —» M to this system is specified by its values on the coordinate axes 23^, which are immediately obtained, via (2.90), from the data (M T^). The last equation in (2.59) implies that this function is a scalar solution of the discrete Moutard equation. Recall that /+ is a solution of the discrete Moutard equation (2.56) with the transformed potential a^~2. The second equation in (2.59) gives the representation (2.89) of a 12 in terms of 9. 2.29. The claim is local, since the property of being Koenigs refers only to the quadrilaterals adjacent to one vertex. Moreover, since the claim is projectively invariant, it is enough to prove it just for one conveniently chosen projection. We choose the projection plane to be the tangent plane 7 at the vertex / of an A-net, so that the points / and f ±i coincide with their projections. We choose the projection to this plane to be orthogonal (so that the center of projection lies at the infinitely remote point in the direction of the normal vector n). For each (i,j) £ {(± 1 , ± 2 )}, let fij denote the orthogonal projection of the point fij to 7: fij = fij ~ c n i

c = (fij ~

/» n ) / (n >n )■

Now we make use of the Lelieuvre representation of the A-net /. One has: f i j —f — riij x rii+ rii x n — ( m j - n ) x n 2, and similarly f i j —f — ( r i i j — n ) x r i j . So, the vector f i j —f is orthogonal to both rii and rij and is therefore parallel to rii x fij- Therefore, f i j — f = arii x n,j — cn. A similar formula holds for M — /, where M is any point on the line ( f f i j ) : (A .7)

M —f =

arii

x

rij

— cn

(since the precise values of a and c in this formula are of no importance for us in this context, we just use the same letters as in the formula for fij —/). We will use the latter result for M being the intersection point of the diagonals ( f f i j ) and ( f i f j ) of the projected quadrilateral (/, f i , f i j , f j ) . The quantity q ( f i f j ) = is given by

/ (M , f j ) / l ( M , f i )

associated to the directed diagonal f i f j

q(fifj) =

-

A

where (A.8)

M — f = n{ f i — /) + A(/j — /),

A + /x=l .

Appendix. Solutions of Selected Exercises

376

We use the Lelieuvre representation on the right-hand side of the latter formula, and compare the result with (A .7): jirii x n + Arij x n = arti x rij — cn.

Considering scalar products of this with n* and r ij , we find: A(rij x n,ni) = —c(n,n*),

/i(n* x n,rij) = —c(n,nj),

and therefore a(TT) = - V =

Clearly, the product of such expressions along a closed path of directed diagonals of quadrilaterals adjacent to / is equal to 1, which, according to Theorem 2.25, proves the claim. A . 2. Solutions of exercises to Chapter 3 3.6. It is sufficient to consider the case N = 3. The four planes in M3 corresponding to the Lorentz unit vectors V{ , Vj , TjVi , T{Vj are orthogonal planes to the corresponding sides of the quadrilateral (/,/*, f i j , f j ) through the midpoints of the sides. Clearly, for a circular quadrilateral, these planes intersect along a line orthogonal to the plane of the quadrilateral through the center of the circle. Therefore, the composition of the reflections in the planes Vi, TiVj coincides with the composition of the reflections in the planes Vj, TjVi. Thus, vi ( nvj ) — ±Vj(rjVi) . The choice of the sign ± is dictated here by equation (3.19). 3.9. Suppose the seven conditions / e ?,

fi £ IP*,

fij e Vij

are fulfilled; we have to show that /123 G CPi23- The eight points /, f i , f i j , and /123 belong to each one of the degenerate quadrics (pairs of planes) y iU ? 2 3 ?

T 3 U T 12,

and therefore they must be the eight associated points. The first seven of them belong also to the plane pair 7 U J)i 23- Since /123 does not lie in !P, it must lie in CP123, as desired. 3.10. According to the previous exercise, the eight vertices of an elementary hexahedron of a discrete A-net are associated points. Therefore, if seven vertices belong to some quadric Q, so does the eighth vertex, as well. 3.17. The orthogonal circles for a pair of faces sharing an edge always intersect in two points and therefore lie on a common sphere. Consider the three orthogonal circles C 12, C23, C 13 for three faces 612, 623^613 sharing a vertex. Draw a sphere through the circles C i 2,C 23. This sphere contains

377

A .3. Solutions of exercises to Chapter 4

the two intersection points of the circle C 13 with C 12, as well as the two intersection points of C 13 with C 23. Thus, four points of the circle C 13 lie on this sphere, and therefore the whole circle does. In the similar manner, one shows that all six face circles lie on the same sphere. Finally, four such spheres attached to four cubes sharing an edge have two points in common: these are the intersection points of all orthogonal circles to the two spheres attached to the endpoints of this edge.

A . 3. Solutions of exercises to Chapter 4 4.3. We have to consider only the second alternative in Lemma 4.13. Thus, suppose that equations (4.24) are fulfilled. As in the first case, we can use one of these relations to fix the gauge transformation in order to maximally simplify all transition matrices Uj. This time the result cannot be made looking the same for all coordinate directions, and this is the reason for restricting these considerations to the case m — 2. It will be convenient for us to change the sign of a 2 at this point (which causes also the corresponding modification of formula (4.25)). We use the equation T1C2 + £2 = t 2t7i - VI

to show the existence of a function k : Z2 —>R/(27rZ) such that 771 = rj( u,u + e\)

=

—k ( u

+ e\) — k( u) ,

€2 = £ ( u , u + e2)

=

- k ( u + e 2) + k( u) .

Then the gauge transformation (4.15) leads to the transition matrices Uj with 771 = 0 and £2 = 0. After that the remaining relation in (4.24), t 2£i

+ f i = 7*1772 - 772,

guarantees the existence of a function : Z 2 —>R/( 47rZ) such that 6 = C ( « , « + ei)

=

^(4>(u + e i) -
% = r) ( u, u + e2)

=

+ e2) + 4> ( u) ) .

Putting these expressions and rji = £2 — 0 into (4.25), we get the Hirota equation (4.35). To finish the proof, we show that in the present case the unit vectors n — $ - 1e 3$ build with necessity an M-net in § 2. This is done similarly to the previous case. The relation 77,12 + n \\ n\ + n2 we want to demonstrate is equivalent to (A.9)

{U' 2 ) - l azU [ + U ^ z U ^ 1 ||

+ U rU ^ a 3.

Appendix. Solutions of Selected Exercises

378

With matrices (4.33), (4.34) we easily compute: (U ^ a s U i + U ^ U ^ 1 = a s U . U ' 1 + L h U ~ la 3

=

^

_ ° p ),

^

_ °^ ) ,

with P

— cos ^

Q

=

cos ^

2 cos ^

2 Due to the equality ^

+ e*^1^ + sin

cos ^ e*^1 — 2 sin ^

2

2

sin ^

('el1]2 + e ~im) ,

sin ^ e- ^'2.

2

cos — cos — (e 1^1 — e*^1) —sin — sin — (e rr}’2 — e- *772) = 0, 2 2 V J 2 2 V / we can represent Q as Q = cos ^

z

cos ^ f e*^1 + e*^1^ —sin — sin — fe M?2 + e ~lT)2^. z

\

/

z

z

\

/

It follows that P ,Q e e*(012+01~02_(W/4 •R,

so that F/Q G R, which proves claim (A.9). 4.4. We can consider the edges f i — f = n\ x n and — f — n x ri2 of the quadrilateral as lying in the tangent plane Tn§2 to §2 at the point n. It is geometrically clear that the angle
ri2 — n2 — (n2,n)n,

thought of as based at n and therefore belonging to T n§2. The lengths of these vectors are readily found from |n'|2 = 1 — ( rij,n) 2 = sin2 aj. Thus, we obtain (n^n^) COS Cp

i / i i / i

|n1||n2|

(^ 1,^ 2) — (p>\, t i ) (ri 2 i t i ) I /i i

/i

lnilln2l

{n\,ri 2 ) — cosai cosa 2 •

•

smaisma2

*

Actually, we have just derived the equation (A .10)

(^ 1^ 2) — cosai cosa 2 —c o s s i n a i sina 2,

which is nothing but the spherical cosine theorem for the spherical triangle with vertices n, n 1, n2. Similarly, (A. 11)

(^ ,^ 12) = cosai cosa 2 —cos ip* sinai sin 0:2.

A .3. Solutions of exercises to Chapter 4

379

It remains to relate the quantities in (A. 10), (A. 11). This is done with the help of the discrete Moutard equation

( n , n i + n 2) . 1 + (n i,n 2)

.

ni2 + n = ------------- - (m + n2),

from which it follows immediately that (1 + (n i,n 2) ) ( l + (ni 2,n)) = (cosai + cosa^)2. Substitute (A. 10), (A. 11) into the last formula; straightforward manipula­ tions lead then to . (A .12)

cos (p + cos ip* sinai sin c*2 — 1 + cos
This is the sought after relation between (p and (p*; it remains to bring it into the form claimed in the exercise, which is a matter of simple trigonometry. One derives from (A. 12): __ 2 { a i + a 2\ tan2 - tan2 — = 1 + C0S(ai + 2 1 + cos(ai —q 2)

^ co^

' 2 ( Ql ~

n

) _ i 1 ~ Kf

(l + « ) 2’ 0

Since |k |< 1, we find: