Differential Geometry

Download Calculus of Variations by Jurgen Jost, Xianqing Li-Jost PDF

By Jurgen Jost, Xianqing Li-Jost

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Example text

We shall later on see that the first conjugate point of a point p £ Sn along a great circle is the antipodal point q of p. 4. 2 Fields of geodesic curves Let M be an embedded, differentiate submanifold of E d , or, more generally, a Riemannian manifold of dimension nf, again of class C 3 . Let MQ be a submanifold of M; this means that Mo itself is a differentiate submanifold of E d , respectively a Riemannian manifold, and that the inclusion i : M 0 c-> M is a differentiate embedding. We assume that MQ has dimension n — 1, and that it is also of class C 3 .

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

Since 7^ depends differentiably on v, and since v G R n , \v\ = 1, is compact, there exists eo > 0 with the property that for all v with \v\ = 1, j v is defined on [—eo, eo]. 5), we then conclude that for any w G E n with M < eo> yw is defined on [—1,1]. 3, eo may be chosen to depend continuously on ZQ. 2 Fields of geodesic curves 47 We now define a map e = eZ0 : {w G Rn : M < e 0 } -+ J7 WH+7^,(1). Then e(0) = z0. 5) = 7«(0) = V. 3. e. e is bijective, and both e and e~l are differentiate) onto a neighbourhood of ZQ G U.

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