The index growth and multiplicity of closed geodesics

In the recent paper [31] of Long and Duan (2009), we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in [31] of Long and Duan (2009) on rational, and in [12] of Duan and Long (2007), [39] of Rademacher (2010), and [40] of Rademacher (2008) on completely non-degenerate closed geodesics on spheres and CP² to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.     

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.     

Fundamental group and contractible closed geodesics

We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c(M) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M)^x. In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for π₁ (M), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.     

F-structure on collapsed orbifolds

In this paper, we prove the existence of nilpotent Killing structures and F-structures on collapsed Riemannian orbifolds. Therefore a sufficiently collapsed orbifold X is the union of orbits, each orbit is an infranil orbifold of positive dimension; in particular, the F-structure provides a decomposition of X into compact flat orbifolds.     

Conjugate points in length spaces

In this paper we extend the concept of a conjugate point in a Riemannian manifold to geodesic spaces. In particular, we introduce symmetric conjugate points and ultimate conjugate points and relate these notions to prior notions developed for more restricted classes of spaces. We generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We close with applications of these new kinds of conjugate points to CBA(κ) spaces: proving both known and new theorems. In particular we prove a Rauch comparison theorem, a Relative Rauch Comparison Theorem, the fact that there are no ultimate conjugate points less than π apart in a CBA(1) space and a few facts concerning closed geodesics. This paper is written to be accessible to students and includes open problems.     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

Automorphism invariant Cartan subgroups and power maps of disconnected groups

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.     

Perturbed closed geodesics are periodic orbits: Index and transversality

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of equals minus its Conley-Zehnder index when viewed as a critical point of and if is trivial. Otherwise a correction term +1 appears. Received: 21 May 2001; in final form: 10 October 2001 / Published online: 4 April 2002     

On the period spectrum of a symplectic mapping

Let M be a compact Riemannian manifold. It has been known for a long time that the singularities of the wave trace, trace(cos √Δt), are located at the periods of the closed geodesics. Do these singularities also contain information about the geometry of M in the neighborhood of a closed geodesic? We prove that the Birkhoff canonical form of the Poincaré map can be determined from the singularities of the wave trace.     

On the new examples of complete noncompact Spin(7)-holonomy metrics

We construct some complete Spin(7)-holonomy Riemannian metrics on the noncompact orbifolds that are ?⁴-bundles with an arbitrary 3-Sasakian spherical fiber M. We prove the existence of the smooth metrics for M = S ⁷ and M = SU(3)/U(1) which were found earlier only numerically.     

Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

Let M ⁿ be a closed Riemannian manifold homotopy equivalent to the product of S ² and an arbitrary (n–2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M ⁿ there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M ⁿ .     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

A weak energy identity and the length of necks for a sequence of Sacks-Uhlenbeck α-harmonic maps

In this paper we discuss the convergence behavior of a sequence of α-harmonic maps uα with Eα(uα)<C from a compact surface (M,g) into a compact Riemannian manifold (N,h) without boundary. Generally, such a sequence converges weakly to a harmonic map in W1,2(M,N) and strongly in C^∞ away from a finite set of points in M. If energy concentration phenomena appears, we show a generalized energy identity and discover a direct convergence relation between the blow-up radius and the parameter α which ensures the energy identity and no-neck property. We show that the necks converge to some geodesics. Moreover, in the case there is only one bubble, a length formula for the neck is given. In addition, we also give an example which shows that the necks contain at least a geodesic of infinite length.     

On Poincaré's isoperimetric problem for simple closed geodesics

We show in the context of integral currents that Poincaré's isoperimetric variational problem for simple closed geodesics on ovaloids has a smooth extremal C without self-intersection. Provided the smooth Riemannian metric on the ovaloid M in question does not deviate too far from constant curvature, we further show that (i) this extremal C is connected and so is the desired non-trivial geodesic of shortest length on M and (ii) C is close (in the sense of Hausdorff distance) to a great circle.     

Integrability of geodesic flows and isospectrality of Riemannian manifolds

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds M and M′ which are isospectral for the Laplace operator on functions and such that M has completely integrable geodesic flow in the sense of Liouville, while M′ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by two maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in M, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for M, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both M and M′ satisfy the so-called Clean Intersection Hypothesis. The author was partially supported by DFG Sonderforschungsbereich 647.     

Non existence of quasi-harmonic spheres

Let M and N be compact Riemannian manifolds. To prove the global existence and convergence of the heat flow for harmonic maps between M and N, it suffices to show the nonexistence of harmonic spheres and nonexistence of quasi-harmonic spheres. In this paper, we prove that, if the universal covering of N admits a nonnegative strictly convex function with polynomial growth, then there are no quasi-harmonic spheres nor harmonic spheres. This generalizes the famous Eells–Sampson’s theorem (Am J Math 86:109–169, [7]).     

Geodesics onS 2 and periodic points of annulus homeomorphisms

Summary We show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric onS ² with positive Gaussian curvature has infinitely many distinct closed geodesics.In this paper we investigate area preserving homeomorphisms of the annulus and their periodic points. The main result is that an area preserving homeomorphism of the annulus which has at least one periodic point (perhaps on the boundary) must in fact have infinitely many interior periodic points.The motivation and main application of this result is the furthering of a program begun by Birkhoff [B] in his book Dynamical Systems. There he shows that for many Riemannian metrics onS ², including those with positive curvature, the problem of finding closed geodesics reduces to finding periodic points of a certain area preserving homeomorphism of the annulus. The annulus map in question can be shown to have a periodic point so our main result above can be applied to show the existence of infinitely many distinct closed geodesics whenever this annulus map exists. This is done in Sect. 4 Other quite different approaches to the problem of finding infinitely many geodesics have been successful in handling the cases which do not reduce to the investigation of an annulus homeomorphism (see [Ba]).Oblatum 20-III-1991 & 6-XI-1991     

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.