Conformal Geometry of Generalized Metric Spaces

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376

PRoS. N. A. S.

ti, . . ., t ranging over H. Hence A has rank tr with respect to H and its rank equation has the symmetric group with respect to K = HQ1, ... .) and is irreducible in K. By applying theorem 1 we have THnOREM 4. Let A be a simple algebra over F expressed by A = B 0 M where M is a total matric algebra of order r2 and B is a normal division algebra in t2 units over its central field R(v) of order s. Then the rank of A is str and the rank function of A with respect to F is irreducible in the field of its coefficients. 1 Dickson, L. E., Algebren und ihre Zahlentheorie, p. 260. 2 Cf. Noether, E., Mathematische Annalen, vol. 78. 3 Wedderburn, Bulletin of the American Mathematical Society, 31 (1925), 11-13. 4

Cf. the author's paper, "On the Group of the Rank Equation, etc.," these PRo-

COUDINGS 14, (1928), No. 12, pp. 906-7.

CONFORMAL GEOMETRY OF GENERALIZED METRIC SPACES BY M. S. KNEBsLMAN* DEPARTMENT OF MATHUMATICS, PRINCETON UNIVERSITY

Communicated February 25, 1929

1. Let V,, be an n-dimensional generalized metric space-one to which there is assigned an absolute scalar differential invariant f(x, dx), f being positively homogeneous of degree two in dxl, . . ., dxT. If f.i =flbdx' and gij =/2fij.J the length of a vector t relative to the element dx is by

definition

t2= g(xdx)t1,

(1.1)

the summation convention for every repeated index being used. I define two metrics f(x, dx) and f'(x, dx) as conformal if the length of an arbitrary vector in the one is proportional to the length of this vector in the other. Let 'p(x, dx) be the factor of proportionality; then from (1.1) we obtain gij = 'pgij

and therefore gij.k = Vgqj.k + (o.kgVj = Vgjk.i + '.igjk

Hence

X'P.k

=

5k4.jX

from which we get by contraction V.k = 0. Hence the factor of proportionality is at most a point fulrction. For convenience we write 1 2a, (1.2) gij e =

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then if gt' is the normalized cofactor of gij in g,,e which is assumed not to vanish, we have g I ij e- 2ai g. (1.3) Let r and r1 be the fundamental affine connections of the two spaces, their components being given' by expressions of the form

rik

=

{jk} +

1 + ti}.kdX+/2 {ik}1dx

{a }j.k

dxadoXa ; (1.4)

{jk} being the Christoffel symbols of the second kind formed out of the g's. When these components are compared we find that

rk= rk- A'0k o

(1.5)

where oac = bal/?)x and A"a = l/2fgia -dxidxa. If Kekz are the components of the affine curvature tensor, we find by means of (1.5) that l~ha i# Kjkl (A Jk 0,al -Ail1 0 ak) + 0a0 A.lh af (A.k Kjl= KkI

(As

h.k).i - a,a(AT,i.j - A-[,kj), (1.6) where a subscript preceded by a comma indicates covariant differentiation with respect to the r's. The conditions of integrability of (1.5) and a., = 0 furnish the complete set of conformal invariants, for the two metrics f and f'. Before outlining the method for obtaining these invariants it may be worth noticing the essential difference between our problem and the corresponding one for a Riemann space. If f is a Riemann metric, the coefficient of 0,a in (1.5) vanishes identically and the algebraic elimination of the ( + ) quantities of ma, carries with it the elimination of 0,a C,, so AI

that the result of elimination is a tensor, -the well-known conformal curvature tensor of Weyl. In the generalized metric space the elimination of ar as gives a conformal invariant which in general is not a tensor while the elimination of the n2 + 2n quantities a a a,o-aa,, and oma if it can be performed, will give a conformal tensor. 2. An invariant which we call a conformal connection may be defined in two ways, each of which involves an additional assumption about the metric f. .The first connection is analogous to the one developed by J. M. Thomas2 and the second one by T. Y. Thomas.3 For Riemann spaces and in fact for generalized metric spaces whose Hessian with respect to dx is a point function these connections are identical, both assumptions being valid for spaces of this sort. Let f be any generalized metric. With T. Y. Thomas we let F f gI - 1/", F being a relative scalar of weight - 2/n. We assume that

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the Hessian of F with respect to dx does not vanish identically. Let x' = (p'(2) be any analytic transformation of coordinates whose jacobian = 2F~ij, (x,x) 0 0; then if Gi1

G-y = (x, )2/n Gaj ugaU7 (2.1) where uE' = ?Xa/)ti and if GCV is the normalized cofactor of G(y in I Gas I - (x, )2/n Gaf Vi

(2.2)

where v ia = 6xa. When we form the affine connection out of the G's (cf. (1.4)) we find that its components Kjk transform according to the law

K~kUs

= K,8Sy~t~k

+ U,,k +

where

=2xa/8jaxk -

,

=

4

A4,klisUi'

(2.3)

1/n a log (x,2)/bs

and

(2.4)

AiS - 1/2 F USS

-

d~idf.

The invariant whose components are Kk we call the conformal connection and if P~ki is the analogue of the curvature tensor formed out of the K's, its law of transformation is

piki

P5U-ukUl vt

pa

(Aiakl-',a/l - A.j l+1/A/k) (A.k - Ah ll'/aol/-+ +k(A-. -A-i/k j) (2.5) +

.

a subscript preceded by a solidus denoting covariant differentiation with respect to the K's. When the expressions for Pjkl + 1/2 Pijk are evaluated we get n(n + 1) 2 equations that may be solved for the n(n + 1) 2 quantities lI'//,a giving expressions of the form

= S'azl [Piki'/2PXt)z4kuZ1 + 1/2Ptjk - (P;.i + + vJ/a/

TVa,6/j~lk +

Uj ,I//j. (2.6)

By means of (2.6) we are able to eliminate higher derivatives of 4i and thus obtain an infinite sequence of conformal invariants which are not tensors in general. These invariants together with Kjk.lh.12. .. /ml/m2... and Gij.ki.k.... constitute a complete set of conformal invariants, the proof of this fact being analogous to that used in the metric equivalence of generalized spaces. 3. By making a further assumption about f(x, dx) we can form a conformal connection which leads to conformal tensors. This assumption is false for spaces whose Hessian is a point function and consequently false for Riemann spaces. That it is not always false can be shown by a simple example.

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From (2.3) we obtain by contraction khk =

Kafki + As(,na + Ahk)

(3.1)

or

Khh

#s (1/2Fp Z).k-

= K au +

We denote the absolute conformal tensor assume that Tkj 0, we may define Sk, by

I

('/2FG%).k

Tk St = Mp.

by

Tk.

If we

(3.2)

Equations (3.1) may then be solved for h giving Ek=-hrr Khrgua -

Hence if we define an invariant whose components are

Ck =

'/2(K',Oix.dxax

- A KfSa).Jk,

(3.3)

we shall find their law of transformation to be

CGkUi

= Cla

U4U+

(3.4)

Thus, if this invariant exists the complete set of conformal tensors is obtained by replacing r~k in the complete set of metric tensors by the corresponding C0.* * NATIONAL RSARCH FoLLow IN MATrsmAnIcs. 1 L. Berwald, Jahr. der Deut. Math. Ver., 1925, 34, pp. 213-20. 2

These PROCOUDINGS, 12, (1926), 389-393. 12, (1926), 352-359.

3 Ibid.,