Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I know from two sources that it is (or at least was) unknown whether there are infinitely many geometrically distinct closed geodesics for every Riemannian metric on $S^3$, the 3-sphere (Weinberger, Computers, Rigidity, and Moduli, 1995, p.101; Berger, A Panoramic View of Riemannian Geometry, 2003, p.461). And from the same two sources, that it is known that there is at least one closed geodesic on any compact Riemannian manifold (Lusternick and Fet). My question is whether or not it is known that there are at least two distinct closed geodesics on $S^3$?

On $S^2$ it is known there are at least three simple closed geodesics (Lusternick and Schnirelmann), and infinitely many periodic geodesics (Bangert, Franks, Hingston). It might help an idea I'm considering if it were known there is more than one closed geodesic on $S^3$. Thanks for pointers!

share|improve this question
    
This question and your last one had no pictures -- is something wrong? –  Allen Knutson Sep 24 '11 at 1:19
    
Ha! Yes, I am slipping! I should have a picture even for a reference request! :-) –  Joseph O'Rourke Sep 24 '11 at 1:46
add comment

1 Answer

up vote 5 down vote accepted

I think the recent work by Huagui Duan and Yiming Long

http://arxiv.org/abs/1008.1458

showed the existence of at least two closed geodesics.

share|improve this answer
    
"The index quasi-periodicity and multiplicity of closed geodesics." Abstract: "In this paper, we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible Finsler (including Riemannian) manifold of dimension not less than 2." Wonderful!! Thanks so much! –  Joseph O'Rourke Sep 17 '11 at 11:23
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.