Conjugate Points in Length Spaces

Details Conjugate Points in Length Spaces Conjugate Points in Length Spaces

2007-09-14

arxiv.org/abs/0709.2331v3

Advances in Mathematics 220 (2009), pp. 791-830.

Mathematics

Metric Geometry

47 pages, 10 figures, added references and comments to prior notions

Scientific paper

In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\rm CBA}(\kappa)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $\pi$ apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems.

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