By Luther Pfahler Eisenhart
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Extra info for An introduction to differential geometry with use of tensor calculus
There exist functions IS 71 7. Covariant Differentiation 41 XiE Cm(U ) such that X = Ei Xia/axi on U. For simplicity let xi(s) = xi(rp(s)) and write Xi instead of (Xi)* ( 4 2, N o . 1). Then (1) is equivalent to Therefore if X , # 0 there exists an integral curve of X through p . Let M be a manifold with an afine connection. Let M and let X , Y be two vector fields on M. Assume X, # 0. 1. p E translation from p to p)(t) with respect to the curve v. Then We shall use the notation introduced above.
T;)) converges to Vz(4 X , ( P * ) , '**, X,(P*>, y, *'*, t:) which represents a geodesic inside V Z 6 J p * )joining p* to q*. a, < < < < < < for 0 t b, 1 i m; in other words y,(t) + y * ( t ) for 0 t b. Since yn(t) E V,(p) (0 t b) it follows that y* contains no points outside the boundary D. Owing to 1) above, we have (p*, q*) E S ; hence S is closed. 11. S is open. I n fact, the same argument as in I shows that the complement ( V6(p)x V,(p))- S is closed. Definition. Let M be a manifold with an affine connection.
Let X and Y be vector fields on M and N,respectively; X and Y are called @-related if d@,(X,) = for all p Y"(d E M. (3) It is easy to see that (3) is equivalent to (Yf) o@ =X(f0 It is convenient to write d@ . 3. (i) Suppose d@ . Xi = for all f E Cm(N). @) = (4) Y or Xo = Y instead of (3). Yi (i = 1 , 2). Then . [XI, X,l [Yl,Yzl. (ii) Suppose @ is a difleomorphism of M onto itself and put f for f E C"(M). Then if X E D1(M), Li@ (fX)"=fox", = @ =f o @-l (Xf)"= x y . Proof. From (4) we have ( Y l ( Y z f ) )o @ = Xl(Y,f o @) = Xl(Xz(f o @)),so (i) follows.