Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Geodesic rigidity in compact nonpositively curved manifolds

Author: Patrick Eberlein
Journal: Trans. Amer. Math. Soc. 268 (1981), 411-443
MSC: Primary 53C20
MathSciNet review: 632536
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature $ K \leqslant 0$. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank $ 1$. Let $ M$, $ M\prime $ denote compact Riemannian manifolds with $ K \leqslant 0$, and let $ \theta :{\pi _1}(M,\,m) \to {\pi _1}(M\prime ,\,m\prime )$ be an isomorphism. Let $ \theta $ also denote the induced bijection on free homotopy classes.

Theorem A. The free homotopy class $ [\alpha ]$ in $ M$ is, respectively, Clifford, bounded or rank $ 1$ if and only if the class $ \theta [\alpha ]$ in $ M\prime $ is of the same type.

Theorem B. If $ M$, $ M\prime $ have dimension $ 3$ and do not have a rank $ 1$ free homotopy class then they have diffeomorphic finite covers of the form $ {S^1} \times {M^2}$. The proofs of Theorems A and B use the fact that $ \theta $ is induced by a homotopy equivalence $ f:(M,\,m) \to (M\prime ,\,m\prime )$.

Theorem C. The manifold $ M$ satisfies the Visibility axiom if and only if $ M\prime $ satisfies the Visibility axiom.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20

Retrieve articles in all journals with MSC: 53C20

Additional Information

PII: S 0002-9947(1981)0632536-5
Keywords: Nonpositive curvature, free homotopy class, periodic geodesic, homotopy equivalence, rank $ 1$, Visibility axiom
Article copyright: © Copyright 1981 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia