Characteristic classes by John Milnor, James D. Stasheff

By John Milnor, James D. Stasheff

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2) < = {Ci;(y))"1 < C'J(y) + ( C ? ( y ) ) " ^ 7 ( y ) , Ce O(n) We now show that to every Euclidean connection of directions is naturally associated a linear connection. Let us consider the connection defined with respect to a covering of M, endowed with orthonormal frames by the matrices n*, such that The linear connection thus defined is evidently independent of the covering chosen. 4) Vgij= - (coji + coy) = 0 It thus follows that the absolute differential of the metric tensor is zero. 4), it is clear that for a covering of M by the neighborhoods endowed with local sections of E(W, g) the matrices of the linear connection are skew-symmetric; on the other hand if 7i"' (UnV) we have > ClJ (y), where C" is an element of the group O(n); for the linear connection envisaged we have the condition of coherence It thus follows that the matrices (oy{!

1) ds = F(x,dx), F is by definition the arc element. 2) s(x o ,xi)= f F(x, x)du (*=|T) F being restricted of degree 1 the integral of the right hand side is independent of the parametric representation chosen for the path l(xo, Xi). 4) n = | £ 8x' and TTO and n\ correspond to the points xo and x\. We call the extremal of the problem of calculus of variations attached to F(x,x). It is a solution of the differential system of the second order. n). v ' where we have put F2(x, v) = 2L(x, v). n). 7) det(<%L)*0 This problem is called positively regular if <%L is a positive definite quadratic form.

8) by v1 and v* successively; in virtue of the homogeneity of F, we have gij (x, v x ) v V = ^ 8jF2 = 2L = F 2 (x, vx). We are thus led to the following definition : Definition . Let M be a differentiable manifold and V(M) the space of non-zero tangent vectors to M. The structure of a Finslerian manifold on M is defined by the data of a function F(x, Vx), positive, positively homogeneous of degree 1 on V(M) leading to a regular problem of the calculus of variationsL In the following we suppose that F leads to a positively regular problem.

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