Thin r-neighborhoods of embedded geodesics with finite length and negative Jacobi operator are strongly convex

Philippe Delanoë

Vol. 264 (2013), No. 2, 307–331

DOI: 10.2140/pjm.2013.264.307

Abstract

In a complete Riemannian manifold, an embedded geodesic γ with finite length and negative Jacobi operator admits an r-neighborhood N_r(γ) with radius r > 0 small enough such that each pair of points of N_r(γ) can be joined by a unique geodesic contained in N_r(γ) where it minimizes length among the piecewise C¹ paths joining its endpoints.

Keywords

geodesic, negative curvature, Fermi chart, tube, convexity

Mathematical Subject Classification 2000

Primary: 34C11, 53C21, 53C22

Secondary: 53B21

Milestones

Received: 22 November 2011

Revised: 18 May 2013

Accepted: 21 May 2013

Published: 28 July 2013