By M. M. Postnikov
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Extra resources for The Variational Theory of Geodesics
In the special case in which N = M, we obtain the result that, in accordance with this definition, an arbitrary vector in MP is tangent to the manifold M at the point p. This explains the name "tangent space" for the space M". • x 111 be an arbitrary system of local coordinates on the manifold M at the point p, and let yl, ... , y" be an arbitrary system of local coordinates on the submanifold N at the point p. Consider the functions xi Lv = xi o t, ... ' x111 IN = xm o t, where t is the embedding mapping N -> M.
On the other hand, if we set ©=a; dx 1, where a1 =a( (a:' )J, we shall obviously obtain on u a form w such that ©P =a. The function p - ©P that assigns to each point p E M the covector ©P obviously satisfies the following 11 smoothness condition": f For an arbitrary vector field XE 6 1 (M), the real junction is smooth on M. (p)=©P (Xp) It is easy to see that, conversely, Any function p - ©,, that assigns to each point p E M a convector E6 1 (p) and satisfies this smoothness condition satisfies some linear differential form w.
Then, the curve y, defined parametrically by the functions x 1 (t), will be an integral curve of the field X that passes through the point p. On the basis of the theorem on the uniqueness of the solution of systems of differential equations, this curve is uniquely defined. Thus, For X p=I= o, there exists exactly one integral curve y(t) of the field X such that y(O) = p. ) We consider, in addition to curves, also smooth n-dimensional surfaces in M , that is, smooth mappings cp of a connected open set OcR" into the manifold M.
The Variational Theory of Geodesics by M. M. Postnikov