Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Geodesic rigidity in compact nonpositively curved manifolds
HTML articles powered by AMS MathViewer

by Patrick Eberlein PDF
Trans. Amer. Math. Soc. 268 (1981), 411-443 Request permission

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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20
  • Retrieve articles in all journals with MSC: 53C20
Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 268 (1981), 411-443
  • MSC: Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0632536-5
  • MathSciNet review: 632536