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By Eisenhart L. P.

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M Proof of Proposition 2. For clarity suppose r = 1 and s = 2. Consider the following telescoping identity (extendible in an obvious way to any r, s): ~ ( ex, , P) - A(e, x , Y ) = A(B - 8, x,P) + A(@,x - x , P) + A(e, x , P - Y). By hypothesis 0 - 8, X - X , and Y - Y all vanish at the point p . Hence, by the lemma, A@, 1, y)(p)= A(8, X , Y)(p). It follows immediately from Proposition 2 that a tensor field A has a value A , at each point p of M , namely, the function E 2:(M) A,:(T,M*)' x (T,M)" + R defined as follows.

X" 0 .. 71, il,. , in). We shall now apply Proposition 42 to make T M a manifold with all such functions as coordinate systems. Thus 4 is a one-to-one function from 71'- '(42) onto the open set 4(%) x R" of R'". 1 Integral Curves 27 Next we show that any two such functions A 1y) x R", then for 1 I i 5 n and t j overlap smoothly. If (a, b) E q(% uiStj-l(a, b) = x'ntj-'(a, b ) = x'q-'(a). Since d/dyi = 2 (dxk/dyi)d/dxk,we also have Thus 4" t j - l is Euclidean smooth. It is easy to check the conditions in Proposition 42 that show T M is a Hausdorff and second countable.

U s ) with ui E F. The usual componentwise definitions of addition and of multiplication by an element of K make V, x ... x V , a module over K, called a direct product (or direct sum if the notation x is replaced by 0). If W is also a module over K , a function A:V1 x ... x K+ w is K-multilinear provided A is K-linear in each slot, that is, for 1 I i I s and u j E vj ( j # i), the function u 4 A ( u , , . . ,ui- 1, is K-linear. 34 0, ui+ 1,. ,us) Tensor Fields 35 If V is a module over K , let V* be the set of all K-linear functions from V to K .

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Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P.

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