On the Equi-projective Geometry of Paths

,592

MA THEMA TICS: T. Y. THOMAS

PROC. N. A. S.

where

br/

2er ~P Z)r

r3

1

2

=j-3- D(t)

-t

=

D(t)

D(t) being arbitrary. p may here be expressed in terms of a p-function whose argument is log r. The theorem which is given above thus reduces the ten elements of form (1) to three fundamental ones. Corresponding theorems are easily developed for use in connection with line elements other than (1), e.g., for those of Kasner's type." * NATIONAL RZSsARCH FELLow in Mathematics. 1 Cf. Kasner, U., Amer. J. Math., 43, 20-28 (1921); Math. Ann., 85, 234-236 (1922). 2 Bateman, H., Electrical and Optical Wave Motion (Cambridge, 1915), p. 31. 3 Kasner, U., these PROCUSDINGS, 11, 95-96 (1925).

ON THE EQUI-PROJECTIVE GEOMETRY OF PA THS BY TRAcy YsRKxs THoMAs* ZUIUCH, SWITZIRLAND

CommUniCated June 29, 1925

1. Introduction.-In this note I wish to give an indication of the method of development of the equi-projective geometry of paths,1 which, although quite analogous to that of the affine geometry, requires some special consideration. The theorems which I have given have their counterpart in the affine geometry of paths; yet the theorem in the last paragraph on complete sets of identities is new even for the affine geometry.2 2. Characterization of Equi-projective Normal Co6rdinates.-Equi-projective normal co6rdinates (z) are completely characterized by the equations

si

za zll

=

O

(2.1)

in terms of an affine connection 5 which is obtained from the projective connection IIH,' in the general (x) coordinate system according to the

equations

$a= = aax (abzaazB +

(2.2)

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MA THEMA TICS: T. Y. THOMAS

Voi,. 11, 1925

and which therefore satisfies the condition

pa(2.3) =vap ~3 Since.the equations (2.2) have the ordinary form of the equations of transformation of the affine connection rap it follows that the transformation of equi-projective invariants to the system of normal coordinates (s) results simply in a replacement of the projective connection II4,O by the connection ,. As a consequence of this and the fact that the (z) coordinates transform by a linear homogeneous equi-transformation, the process of extension of invariants of the manifold involving repeated differentiation of the invariant and evaluation at the origin of normal coordinates as developed in my thesis- is valid for the equi-projective geometry of paths. That this process of extension may be applied to general tensors is evident. Contracting the indices in (2.2) we have the equations Va

=

; A =

a

x/zI

(2.4)

which we shall use in the following paragraph. 3. Equi-projective Plane Manifolds.-A manifold is equi-projective plane if it can be mapped on a Euclidean space by an equi-transformation in such a way that the paths of the manifold go into the straight lines of the Euclidean space. An obvious condition that the manifold be equiprojective plane is that the equi-projective normal tensor ?i, vanishes identically. To see that this condition is also sufficient consider the expansion p=

o

+

21

Ia#zZ

+

*

(3.1)

about the origin of the normal co6rdinate system (z). It may be shown that the vanishing of the equi-tensor ['IZ necessitates the vanishing of all the other equi-tensor coefficients in the series (3.1).4 Reference to the equations (2.4) then shows that the transformation to normal coordinates (z) is an equi-transformation. Hence A necessary and sufficient condition Jor the manifold to be equi-projective plane is that the equi-projective normal tensor Kt p, vanishes identically. 4. Replacement Theorem.-The process of transforming any equi-pro-

jective invariant

E.1::

I$;I/x;..;b,8t..d° (4.1) to the system of equi-projective normal coordinates (z) and evaluating at the origin enables us to see immediately that E may be given the form *

This gis us tefepl t (reductio) This gives us the following replacement (reduction) theorem.

(4.2)

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MA THEMA TICS: T. Y. THOMAS

PRtoe. N. A. S.

Any equi-projective invariant (4.1) can be put into the form (4.2) by replacing the Us by zero and their derivatives by the components of the corresponding equi-projective normal tensor W. In the sense of this theorem we may say that the equi-projective normal tensors 2l constitute a complete set of invariants of the manifold. 5. Complete Sets of Identities.-From the equations (2.1), (2.3) and (3.1) we obtain the identities satisfied by the- equi-projective normal tensors, namely

s az ...C) = 0 (5.2) where (G... v) in '(5.1) denotes any permutation of the indices ('y...a), and S in (5.2) denotes the Sgum of the terms obtainable from the one in the parenthesis which are not identical by (5.1). The identities (5.1) and (5.2) constitute a complete set of identities for (See note 2.) To prove the equi-projective normal tensor F . this consider a sequence of sets of numbers .0pS;

W1107 8; 21'apt Be

...

(5.3)

chosen so as to satisfy the algebraic conditions (5.1) and (5.2) and also so that the series (3.1) converges but otherwise quite arbitrary. The fiuctions !$'P defined by (3.1) in which the coefficients W are the sets of numbers of the sequence (5.3) then satisfy the equations (2.1) which characterize the variables (z) as a set of equi-projective normal coordinates. Hence The identities (5.1) and (5.2) constitute a complete set of identities for the equi-projective normal tensor !K'ap..... Thus any set of identities satisfied by the equi-projectiVe normal tensor can be deduced by algebraic processes 'from the identities of the complete set (5.1) and (5.2). The sets of identities (5.1) and (5.2) constitute all the algebraically independent sets of identities of the complete set of invaiants K which exist in the manifold.

* NATIONAL RUSUARCc FVniaow in Mathematical Physics. 1T. Y. Thomas, these PROConDiNGS, 11, p. 199. 2The idea of a complete set of identities of an invariant of the manifold has been introduced.by me in a paper on the identities of affinely connected' manifolds which will appear in the Math. Zeitschrift. By this is meant a set of algebraically independent identities of the invariant constituting all the algebraic conditions on the invariant so that any identity satisfied by the invariant can be deduced from the identities of the complete set by algebraic processes. - In the above paper I have treated in detail the cotnplete sets of affine and metric (Riemann) invariants of the manifold. ' 0. Veblen and T. Y. Thomas, Trans. Amer. Math. Soc., 25, 1923, p. 551. 'The theorem involved is analogous to one given by Vermeil for the case of the Riemann geometry. See, H. Vermeil, Math. Ann., 79 (1918), p. 289.