Katz N.M.'s A conjecture in arithmetic theory of differential equations PDF

By Katz N.M.

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Inasmuch as Gi = 0 for t = 0, we obtain I*( Xiaiwi) = 0. Moreover z* so, using Y* = I*o@* (pztj = 1, the lemma follows. 50 ELEMENTARY DIFFERENTIAL GEOMETRY [Ch. I We shall now introduce the Riemannian metric on the Riemannian manifold M . Let t -+ y ( t ) ( a t p) be a curve segment in M . T h e arc length of y is defined by < < L(Y) = J B {gyw(+(% i,(t))>”2 dt. (3) It is clear from (3) that two curve segments which are the same except for a change of parameter have the same arc length. ” It will also be convenient not always to distinguish between two curves which coincide after a change of parameter.

K It is clear that the forms wii determine the functions on N , and thereby the connectionv. On the other hand, as the next theorem shows, the forms wii are described by the torsion and curvature tensor fields. 1 (the structural equations of Cartan). Both sides of ( I ) represent a 2-form on N,. We apply both sides of that equation to (Xi, X,)and evaluate by means of the rules (4) and (9) 9 81 8. The Structural Equations 45 in $ 2. If we define the functions cijk by [ X j , X,] left-hand side of (1) is dWi(Xj,X , ) = + { X j .

2. Let p , q be two points in M and y a curve segment from p to q. The parallelism r with respect to y induces an isomorphism of M p onto Mq. Proof. Without loss of generality we may assume that y has no double points and lies in a coordinate neighborhood U . , x ? ~ } be a system of coordinates on U. Suppose the curve segment y is given t b) such that y ( a ) = p , y(b) = q. by the mapping t + y(t) ( a As before we put xi(t) = xi(y(t)), ( a t b), (1 i m). Consider the system (2). , y,) satisfy the system (2).

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A conjecture in arithmetic theory of differential equations by Katz N.M.

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