Vol. 302, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 310: 1
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics

Ming Xu

Vol. 302 (2019), No. 1, 353–370

In this paper, we consider a Finsler sphere (M,F) = (Sn,F) with dimension n > 1 and flag curvature K 1. The action of the connected isometry group G = Io(M,F) on M, together with the action of T = S1 shifting the parameter t of the closed curve c(t), define an action of Ĝ = G × T on the free loop space ΛM of M. In particular, for each closed geodesic, we have a Ĝ-orbit of closed geodesics. We assume the Finsler sphere (M,F) described above has only finite orbits of prime closed geodesics. Our main theorem claims that, if the subgroup H of all isometries preserving each close geodesic is of dimension m, then there exists m geometrically distinct orbits i of prime closed geodesics, such that for each i, the union Bi of geodesics in i is a totally geodesic submanifold in (M,F) with a nontrivial Ho-action. This theorem generalizes and slightly refines the one in a previous work, which only discussed the case of finite prime closed geodesics. At the end, we show that, assuming certain generic conditions, the Katok metrics, i.e., the Randers metrics on spheres with K 1, provide examples with the sharp estimate for our main theorem.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Katok metric, Randers sphere, constant flag curvature, orbit of closed geodesics, totally geodesic submanifold, fixed point set
Mathematical Subject Classification 2010
Primary: 22E46, 53C22, 53C60
Received: 29 April 2018
Revised: 27 October 2018
Accepted: 22 March 2019
Published: 5 November 2019
Ming Xu
School of Mathematical Sciences
Capital Normal University