Closed Geodesics Research Articles

The Non-contractibility of Closed Geodesics on Finsler ℝPn

The Non-contractibility of Closed Geodesics on Finsler ℝPn

On the existence of closed geodesics on 2-orbifolds

We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.

Morse concavity for closed geodesics

In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.

Morse Indices of Closed Geodesics on Katok's 2-Spheres

Abstract In this paper, we compute precisely Morse indices of iterates of the two closed geodesics on the 2-sphere with Katok’s metric. This information is used to give a direct proof for the Poincaré series of the free loop space on S2.

The Length of a Shortest Closed Geodesic on a Two-Dimensional Sphere and Coverings by Metric Balls

In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold M diffeomorphic to the standard two-dimensional sphere. The first result is that the length of a shortest closed geodesic l(M) is bounded from above by 4r , where r is the radius of M . (In particular that means that l(M) is bounded from above by 2d, when M can be covered by a ball of radius d/2, where d is the diameter of M.) The second result is that l(M) is bounded from above by 2( max{r 1,r 2}+r 1+r 2), when M can be covered by two closed metric balls of radii r 1,r 2 respectively. For example, if r 1 = r 2= d/2 , thenl(M)≤ 3d. The third result is that l(M)≤ 2(max{r 1,r 2 r 3}+r 1+r 2+r 3), when M can be covered by three closed metric balls of radii r 1,r 2,r 3. Finally, we present an estimate for l(M) in terms of radii of k metric balls covering M, where k ≥ 3, when these balls have a special configuration.

The Relation of the Morse Index of Closed Geodesics with the Maslov–type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic.

Filling Radius and Short Closed Geodesics of the $2$-Sphere

Nous montrons que la longueur de la plus courte courbe non triviale parmi les geodesiques simples fermees d'indice zero ou un et les geodesiques en huit d'indice nul fournit une minoration sur l'aire et le diametre des deux-spheres riemanniennes.

Iterated index formulae for closed geodesics with applications

In this paper, various precise iteration equalities and inequalities of Morse indices for the closed geodesics are established. As applications of these formulae, multiplicity results of closed geodesics on some Riemannian manifolds are proved.

THE BETTI NUMBERS OF THE FREE LOOP SPACE OF A CONNECTED SUM

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

Equidistribution of holonomy about closed geodesics

Equidistribution of holonomy about closed geodesics

Wave Invariants for Non-degenerate Closed Geodesics

This paper generalizes the methods and results of our article xxx.lanl.gov math.SP/0002036 from elliptic to general non-degenerate closed geodesics. The main purpose is to introduce a quantum Birkhoff normal form of the Laplacian at a general non-degenerate closed geodesic in the sense of V.Guillemin. Guillemin proved that the coefficients of the normal form at an elliptic closed geodesic could be determined from the wave invariants of this geodesic. We give a new proof, and extend its range to any non-degenerate closed geodesic.

Asymptotic Auto-Correlation for Closed Geodesics

Asymptotic Auto-Correlation for Closed Geodesics

The Fadell-Rabinowitz Index and Closed Geodesics

The Fadell-Rabinowitz Index and Closed Geodesics

The Picard Group, Closed Geodesics, and Zeta Functions

The Picard Group, Closed Geodesics, and Zeta Functions

Closed geodesics on non-simply-connected manifolds

Closed geodesics on non-simply-connected manifolds

Homology and Closed Geodesics in a Compact Negatively Curved Surface

Homology and Closed Geodesics in a Compact Negatively Curved Surface

On the number of closed geodesics on the 2-torus

On the number of closed geodesics on the 2-torus

Area and the length of the shortest closed geodesic

Area and the length of the shortest closed geodesic

Homology of closed geodesics in a negatively curved manifold

Soit M une variete de Riemann compacte dont le flot geodesique sur le fibre tangent unite est de type Anosov. Si on note par N(x, α) le nombre de geodesiques premieres P dans M dont la classe d'homologie est un α donne et de longueur l(p)≤x, alors le taux de croissance exponentiel, lim x→∞ x a1 log N(x, α) est egal a l'entropie topologique h du flot geodesique (>0)

One-sided closed geodesics on surfaces.

One-sided closed geodesics on surfaces.