On the number of regions formed by arrangements of closed geodesics on flat surfaces

Mathematical Notes -     

On the volume of certain manifolds with closed geodesics

It is proved that the ratio of the volume of a Riemannian manifold of cohomology typeCP ² orCP ³ for which all geodesics are closed and of length 2 to the volume of the standard sphere assumes a fixed value.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 102–106     

On the lengths of closed geodesics on a two-sphere

Let be an isolated closed geodesic of length on a compact Riemannian manifold which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that has a sequence , , of prime closed geodesics of length where and . The hypotheses hold in particular when is a two-sphere and the ``shortest' Lusternik-Schnirelmann closed geodesic is isolated and ``nonrotating'.

     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \(\left[ {\frac{{n + 1}}{2}} \right]\) or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S³, F) if any prime closed geodesic has non-zero Morse index.     

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001     

On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and \(\sigma \) be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair \((g,\sigma )\) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle \(T^*E\) of a suitable \(S^1\)-bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We thenstudy the relation between the stability property of energy hypersurfacesin \((T^*M,dp\wedge dq+\pi ^*\sigma )\) and of the corresponding codimension2 coisotropic submanifolds in \((T^*E,dp\wedge dq)\) arising via symplecticreduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545–570, 2016) in this setting.     

Qualitative counting closed geodesics

Geometriae Dedicata - We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We...     

Asymptotic distribution of closed geodesics

In this paper the distribution of closed geodesics on surfaces of constant negative curvature are studied from a dynamical viewpoint. Asymptotic estimates are derived independently of the work of Selberg or Margulis, or the work of Bowen on Axiom A flows.     

On the average indices of closed geodesics on positively curved Finsler spheres

In this paper, we prove that on every Finsler n-sphere (S ⁿ , F) for n ≥  6 with reversibility λ and flag curvature K satisfying ${(\frac{\lambda}{\lambda+1})^2 \, < \, K \, \le \, 1}$ , either there exist infinitely many prime closed geodesics or there exist ${[\frac{n}{2}]-2}$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n?3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite.     

On the existence of closed timelike geodesics in compact spacetimes. II

 The purpose of this paper is on the one hand to extend and generalize, in terms of Clifford translations, some results in a previous paper (Math. Z. 239 (2002), 277–291) concerning the existence of closed timelike geodesics in compact spacetimes, and on the other hand to prove that a compact flat spacetime (M, g) contains a closed timelike geodesic if and only if the fundamental group π₁(M) contains a non-trivial timelike translation. Received: 22 January 2002; in final form: 12 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 53C50, 53C22.     

Holomorphic torsion and closed geodesics

Extending the rank-one case by David Fried, the holomorphic torsion of a compact locally symmetric manifold is expressed as a special value of a zeta function built on geometric data (closed geodesics and their monodromy) of the manifold.