By Jurgen Jost, Xianqing Li-Jost

ISBN-10: 0521642035

ISBN-13: 9780521642033

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Let 7(f), h < t < t2 be another curve in U with 7(^i) = 7 ( - e ) , 7(t 2 ) = 7(e). 3) 46 Geodesic curves with strict inequality, unless 7 is a reparameterization of 7. 4) / since # n n = 1, ^ n = 0 for i = 1 , . . 1(iv), rt<2 1 ^ (t)| * > 7 n (t 2 ) - 7 n (*i) = 7"(e) - 7 n (~e) = L( 7 ). The first inequality is strict, unless 7* is constant for i = 1 , . . , n — 1, and the second one is strict, unless jn(t) is monotonic. d. Following Weierstrafi, we say that the geodesies 7 (£) = {x{ = a,xn = t,-c

3. 21) has a unique solution y(t) with 7(0) = z , 7(0) = v for t G [—€, e] and some e > 0. Moreover, ^(t) depends differentiably on the initial values z, v. 2. 21) are called geodesies on M. 1. , x n+1 ) G M n + \ Y^ (xif = l \ C Mn+1 is a differentiable manifold of dimension n. In order to construct local charts, we put fix := 5 " \ { ( 0 , 0 , . . ,0,1)} ,n2:=Sn\{(0,0,--,0,-1)} 40 Geodesic curves and define 01 : fix -+ E n , g2 : fi2 -+ Mn as and 92(lV x - ° + , ) = (r^T i^«) (<7I and 02 are the stereographic projections from the south and north pole, respectively).

E. e. e. the reparameterized curve c = cor is parameterized by arc-length. 1. Let c : [0, L(c)] —• Rd be a curve parameterized on [0,L(c)]. e. we keep the interval of definition fixed, namely [0, L(c)]), the parameterization by arc-length leads to the smallest energy. 12) whereas for any other parameterization of c on the same interval, L(c) < 2E(c). 13) We now return to those curves c that are confined to lie on M, in order to discover a third invariance. 6) for its energy. 5) has to be independent of them, too.

### Calculus of Variations by Jurgen Jost, Xianqing Li-Jost

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