Multiple closed geodesics on Riemannian 3-spheres

Abstract

In this paper, we prove that for every Riemannian Q-homological 3-sphere (M, g) with injectivity radius inj(M) ≥ π and the sectional curvature K satisfying 1 16 < K ≤ 1 there exist at least two geometrically distinct closed geodesics. Mathematics Subject Classification (2000) 58E10 · 53C22 · 37C27

Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

“…Note that {w h } is a bounded sequence by the second inequality of (3.11). We now use the method in the proof of Theorem 5.4 of [27] to estimate M q (−1) = q h=0 w h (−1) h . By (6.1) and (1.17 ) ≤ q .…”

“…(6.1)Note that {w h } is a bounded sequence by the second inequality of (3.11). We now use the method in the proof of Theorem 5.4 of[27] to estimateM q (−1) = h k l (c 2m−# s ∈ N ∪ {0} | h − i(c 2m−1+sn j j l+i(c j ) k l (c 2m−# s ∈ N ∪ {0} | l + i(c 2m−1+sn j j ) ≤ q .On the one hand, we have# s ∈ N ∪ {0} | l + i(c 2m−1+sn j j ) ≤ q = # s ∈ N ∪ {0} | l + i(c 2m−1+sn j j ) ≤ q, |i(c 2m−1+sn j j ) − (2m − 1 + sn j )î(c j )| ≤ 2n ≤ # s ∈ N ∪ {0} | 0 ≤ (2m − 1 + sn j )î(c j ) ≤ q − l + 2n = # s ∈ N ∪ {0} | 0 ≤ s ≤ q − l + 2n − (2m − 1)î(c j ) n jî (c j )≤ q − l + 2n n jî (c j ) + 1.…”

The index growth and multiplicity of closed geodesics

“…[BTZ1], [BTZ2], [DuL1], [DuL2], [LoW1], [Rad3], [Rad4], [Rad5], [Rad6]), except the Theorem C below proved recently in [LoD1] for the 3-dimensional case and [DuL3] for the 4-dimensional case.…”

Stability of closed geodesics on Finsler 2-spheres

“…(Cf. Satz 6.13 of [26], Lemma 3.10 of [3], and Proposition 2.6 of [22].) Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso).…”