(Dex7111) Differential Geometry Curves - Surfaces by Unknown

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Language: eng
Format: epub

284 7. Spaces of Constant Curvature which holds since the endpoints are taken to be fixed. Then we have (VTX VtX) - (R(T, X)X, T) ) dt. S 0 q >0 <0 The integrand is thus strictly positive, and it follows that the integral is as well. Note that (R(T,X)X,T) K(X'T) ~ —J^x)— is exactly the sectional curvature in the X, T-plane, which is by as- assumption strictly negative. Therefore we have -^r | _.. > 0 and ^L-o = ^ ^or a^ sucn X, hence for every one-parameter family in any direction. Consequently, the function has a strict local min- minimum at c. This implies that the neighboring curves (in this sense) are strictly longer than c. In the special case of hyperbolic space Hn, in fact every geodesic from p to q is strictly shorter than any other curve joining the two points. ? Remark: For a curve c: [a,b] —> M the quantity rb E(c) := / (T, T)dt J a is called the energy functional of c. Under the same assumptions as in 7.8, one has ^ = 2^f , so that the critical curves with respect to L coincide with those with respect to E (up to the parametriza- tion). In local coordinates we have the equation for a geodesic VcC = 0 «=> ck + ^ clrJT^ =0 for k = 1,.. ., n. The local existence of geodesies follows from this (cf. 4.12 and 5.18): Theorem. (Existence of geodesies) At a given point p ? M and for a given vector V ? TpM, (V, V) — 1, there exists locally a unique geodesic Cy with Cy @) = p and

7B Geodesies and Jacobi fields 285 If one considers the set of all geodesies in all directions passing through some fixed point, one is lead to the exponential mapping. Recall the definition of this from 5.19. 7.10. Definition. (Exponential mapping) For a fixed point p ? M let Cy denote the uniquely determined geodesic parametrized by arc length through p in the direction of the unit vector V. In a certain neighborhood U of 0 € TpM, the following map is well defined: Here we have chosen the parameter in such a way that (p, 0) i—»p. This mapping is called the exponential mapping at the point p, and it is denoted by expp: U —> M. For variable points p one can similarly define exp: U —¦ M by setting exp(p,tV) = expp(tV) = Cy (t), where U is an open set in the tangent bundle TM, for example U = {(p,X) | ||X|| < e} for an appropriately chosen e > 0, if M is compact and g is positive definite. Remark: expp maps the lines through the origin of the tangent space to geodesies, and this is done in an isometric manner. In all directions perpendicular to the geodesies through p the map expp is in general not isometric. In what follows it will be important to precisely describe how far this is from being isometric, in particular in the case of constant sectional curvature.

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