Normal coordinates in the geometry of paths

42

MATHEMATICS: H. LEVY

PRoc. N. A.-S.

NORMAL CO0RDINA TES LIN THE GEOMETRY OF PA THS BY HARRY LInVY DEPARTMENT Or MATHEMATICS, UNIVERSITY OF ILLINOIS Communicated May 28, 1930

1. Riemann' first proved that in a metric space characterized by the linear element2 (1.1) ds2 = giadxtdxl -it is always possible to choose a coordinate system first so that the first derivatives of the g's are zero at any preassigned point and second -so that the geodesics through that point have linear equations. Fermi3 extended Riemann's results, establishing the existence of a coordinate system in which the g's are stationary along any preassigned curve, but in such a coordinate system the geodesics will be given by power series in the arc s in which the coefficients of only the quadratic terms are zero. Veblen4 and Eisenhart5 extended Riemann's and Fermi's results, respectively, to non-metric characterized by the paths d2xs dx'e dxk (1.2) ds2 +'k ds ds = 0.

Eisenhart's work has been carried somewhat further by Whitehead and Williams.6 But aside from Veblen no one has found any new coordinate systems in which is retained the vital property of Riemann's results, namely, that geodesics have linear equations. We propose to do this. The main purpose of this paper is to prove the following theorem: Let Vm be an arbitrary m-dimensional manifold in the space characterized by equations (1.2) and let X(,)i (i = 1, 2, ... n) be the components of n - m (a = m + 1, m + 2, ... n) arbitrary directions defined at points of V/rn but none of which lie in V.; then there exist (infinitely many) coordinate systems such that the paths through the points of V,, in directions linearly dependent on the n - m preassigned directions X(,)' have linear

equations. If the space (1.2) is Riemannian, the following somewhat special form of the theorem seems most useful: In any Riemannian space there exist coordinate systems in which the geodesics through the points of a preassigned subspace in the directions orthogonal to that subspace have linear equations. We observe that this theorem may, for m = 0 be regarded as identical with Riemann's. 2. The proof is direct. Let V. be given parametrically by the equations

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VOL. 16, 1930

493

U2, ... ) i=1,2,... n, xi== 1,i(Ul, and let Xi be any direction linearly dependent on the X(r)'S2 Xs= Aa>(^)i

(2.2)

The path through a point x in a direction X is given by4 i + Xis - 1/2rikXi Xks2 Xi 1/3! riJklXXXS where the r's are evaluated at the point x. We write uV = As a = m+ 1, ... y, so that2 X's =

X,

(2.1)

(

(2.4)

S

and xi

xi

+

uVX(s)_ 1/2 PjkX() X(r)

U

-

(2.5)

If we regard the x's in these equations as the functions of u', u2, ... given by (2.1) the X's are then defined as independent functions of the n u's with single valued inverses in the neighborhood of Vm and hence the u's may be regarded as a new set of coordinates. The path (2.3) determined by the initial conditions Xi = X (uO, uO,

X= A`(uo)

.. U )

X(G)i(uo)

that is, by a point (ul, u2, ... u'm') of Vm and a direction there, must satisfy the conditions Ua ka

a= ks

(2.6)

where the k's are constants. We shall speak of coordinates of this type as normal with respect to Vm. 3. Let the coordinates (u', u2, ... u') be normal with respect to the Vm given by u =0 af = m + 1, ... nP (3.1) and let us introduce new coordinates usi defined by the equations a = aa(ul, u2, ... U ) (3.2) (3.3) fi = aauT where the a's are constants and the W's are functionally independent. It follows that the ft's are likewise normal with respect to VM. With the aid of (3.2) and (2.5) we note that when we make a general trans-

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494

PROC. N. A. S.

formation of the parameters ul... um of Vm the normal coordinates will undergo the transformations given by (3.2) and ja = U

a =

m + 1, ... n,

while if we make a general transformation of the space coordinates or if we replace the n - m directions 'X()i associated with the normal coordinates by a new set of X's the transformation of the normal coordinates will be given by (3.3) and ft = Ua, (a = 1, 2, ..., m). 4. Let Luk be the coefficients of the linear connection determined by (1.2) in terms of coordinates normal with respect to a given Vm. Then (2.6) must be a solution of

d2u' + L due duk ds2

(4.1)

ds ds

whence it follows that

La7kakt

= O.

(4.2)

Multiplying by S2 we find by means (2.6) and the equations immediately preceding that the relations (4.3) LVUT"UT = 0 i = 1,2, ..., n hold identically throughout the space. Likewise from (4.2), since the k's are arbitrary, it follows that O

L'

TO

i = 1, 2, ... nn, ,r= m + 1, ..., n,

(4.4)

for all points of Vm. If we differentiate (4.3) partially with regard to uT we obtain 7uoruat +

2Laruo = 0,

whence it follows that the relations 6VI 7

uTuT

=

0

(4.5)

hold throughout the space; again, since the k's in (2.6) are arbitrary that the relations p UT = 0, (4.6) where P stands for the sum obtained by permuting the indices 7r, a, X cyclically, hold for all points of V.. We can continue this process and obtain a sequency of identities of the types of (4.3) and (4.5) which for m = 0 reduce to those obtained by Veblen.4

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5. Let us suppose our space is Riemannian with linear element (1.1), that in it we are given a non-minimal subspace Vm defined by (3.1) and that the linear element of Vm is given by2

ds2 =

daBduado,

where a bar, as over the g, denotes that the function barred is evaluated on V", i.e., for uf = 0 (a = m + 1, . .. n). We assume, moreover, that the coordinates uis are normal with respect to Vm and that the X's associated with normal coordinates have been taken as a set of independent mutually orthogonal normals. Since the curves of parameters uf and ush, respectively, through a point of Vm are orthogonal, it follows that

gA

=

0

-n l,;Bn(51

Moreover we may, by a proper choice of u', make

g= 1- 1.

(5.2)

(o~a~ Q(¢)B=

(5.3)

If we define the quantities Q 22 bug

it follows by a direct computation using (5.1), (5.2) and (4.4) that these U's are the coefficients of the second fundamental forms of V,. A similar computation shows that the functions ,u(,,). defined by

(5.4)

g= 1 (8U)

are the coefficients of the linear forms associated by Voss with a subspace.7 This interpretation of the O's is identical with Bianchi's for m = n - 1,8 and one could by following his method for that case obtain the GaussCodazzi equations for a general subspace directly by means of (5.3) and (5.4). 6. Let m = 1 so that Vm is a curve C. We assume that space is Riemannian referred to coordinates normal with respect to C, and that the X's of §1 are the n- 1 principal normals to C.9 The Frenet equations for C9 Aijdud ds

alA Pa-l

~

al(~~

(6.1)

PaE

where the i/p's are the curvatures of C, reduce in this case by means of (4.4), (5.1) and (5.2) to

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PHYSICS: W. A. MARRISON

La'+=

4+L Lo

-

al

_

Pa

PRoc. N. A. S.

(6.2)

Pa-1

so that 1 (

1gle+ 1 _

gle

(6.4)

or PU

(f= + lao) 1

(6.3)

that is, the curvatures of a curve correspond to the /h's of a subspace. The analogues of the O's are, of course, zero. 'Gesammelte Werke, 1876, p. 261. The Latin indices h, i, j,. . . range through the values 1, 2, ... n, the Greek a, jS, y.. . through 1, 2, . .. m, and the Greek or, a, T, . . . through m + 1, m + 2, . .. n. Repetition of an index indicates the sum obtained by allowing that index to take on all values of its range. 3 Rend. Lincei, Rome, 311, 21, 51 (1922). 4These PROCUEDINGS, 8, 192-197 (1922). 5"Non-Riemannian Geometry," Amer. Math. Soc. Colloq. Publ. (1927), p. 64. 6 Ann. Math., 312, 151 (1930). 7L. P. Eisenhart, Riemannian Geometry, Princeton, pp. 159-163. 8"Lezioni di Geometria Differenziale," Bologna, 2, 450-455 (1924). Riemannian Geometry, pp. 106-107. 2

THE CRYSTAL CLOCK By W. A. MARRISON BELL TZLZPHONZ LABORATORIZS, Nnw YORK CITY Read before the Academy, April 29, 1930

The crystal clock is a relatively new device for keeping accurate time. It consists essentially of a generator of constant frequency controlled by a resonator made of quartz crystal, with suitable means for producing continuous rotation controlled by it to operate time indicating and related mechanisms. A crystal clock of this sort has been set up in the Bell Telephone Laboratories and has been operating over a considerable period. The apparatus was designed especially as a reference standard of frequency for the Bell System, but it was recognized that it might also serve as a reference standard of time, if developed for that purpose. As a matter of fact, since time interval and frequency are so closely related, it would be very