By M. M. Postnikov
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Extra resources for The Variational Theory of Geodesics
Example text
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.