The Variational Theory of Geodesics by M. M. Postnikov

By M. M. Postnikov

Compact, self-contained textual content by way of famous theorist offers the main primary elements of recent differential geometry in addition to the elemental instruments required for the learn of Morse conception. complicated therapy; analytical instead of topological elements of Morse idea emphasised. Contents: 1. delicate Manifolds. 2. areas of Affine Connection. three. Riemannian areas. four. The Variational houses of Geodesics. Appendix Focal issues. five. a discount Theorem. Index. Unabridged republication of the 1967 edition.

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Ym-1), in some neighborhood of the point q; that is, f depends smoothly on the functions y 1, .... ym-1 close to the point q. Finally, the mapping YJ : V---+ Rm- i defined by the formula YJ(q)=(yl(q), ... , ym-1(q)), qEV, is connected with the coordinate homeomorphism formula £(q)=(a, YJ(q)), s: U - R111 by the qEV. From this it follows that this mapping maps the neighborhood V homeomorphically onto the intersection of the set £U and the hyperplane t 1 =a. This proves that the functions y 1, ••• , ym-1 are local coordinates of the premanifold [cp =a) in the neighborhood V of the point p.

Vectors We shall call the linear mapping introduced above XP: 6 (p)- R the vector of the field X at the point p. We shall call the linear mapping A: 6(p)-R a vector of the manifold M at the point p if, in some neighborhood of the point p (or, equivalently, on the entire manifold M), there exists a vector field X such that A = X p• It turns out that A linear mapping A ~ 6 (p)- R is a vector of the manifold Mat a point P if and only if A(fg)=A/ · g(p)+ /(p) ·Ag for all f and g in 6 (p). The necessity of this condition follows immediately from relation (2) of section 5.

Ym denote an arbitrary basis of the module 6 1 (U) and let (:) 1 , ••. , Sm denote the corresponding basis of the module 6 1 (U) [cf. section 7]. Then, for arbitrary forms w1••••• w' E6 1 (U) and arbritary fields TE 6~ (M) on some coordinate neighborhood X 1. . . X,E6 1 (U) T (w 1, ••• , w'; X 1, ••• , X,) = kl ... k, I = T l I ' ' ' IS (l)k I r Xl1 " " ' Wk r I ' ' " 0 Xis s n U ' (2) where T Ik1 ... k, I I ... wk s == T(Sk1 , ... • , yI ) s I (3) (for l = I, ... , r and k = I, .... m) and the X 1.

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