Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Minimal rays on closed surfaces

Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.     

Short separating geodesics for multiply connected domains

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.     

Consequences of contractible geodesics on surfaces

The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.

     

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.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L ^6g+2b+2c–6. This answers a long-standing open question.     

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.     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

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.     

Geodesic nets on the 2-sphere

In this paper we introduce the concept of a geodesic net, an idea which plays the role among graphs that geodesics play among simple closed curves. We establish the existence of specific geodesic nets on the 2-sphere in certain cases.

     

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     

Bumpy metrics on spheres and minimal index growth

The existence of two geometrically distinct closed geodesics on an n-dimensional sphere \(S^n\) with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some \(N \in \mathbb {N}\) all closed geodesics of index \(\le \)N of a non-reversible and bumpy Finsler metric on \(S^n\) are geometrically equivalent to the closed geodesic c, then there is a covering \(c^r\) of minimal index growth, i.e.,

$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$

for all \(m \ge 1\) with \(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for \(N =\infty \) as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large \(L>0\), we obtain on \(S^2\) a metric of positive flag curvature carrying only two closed geodesics of length \(<L\) which do not intersect.

     

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.     

Closed geodesics on pairs of pants

We study the non-simple closed geodesics of the Riemann surfaces of signature (0, 3). In the aim of classifying them, we define one parameter: the number of strings. We show that for a given number of strings, a minimal geodesic exists; we then give its representation and its length which depends on the boundary geodesics.     

THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

This paper studios the existence of closed geodesics in the homotopy class of a given closed curve. Let M be a complete Riemannian manifold without boundary, f∈C~1(S~1, M). Look at S~1 as [0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ(f_i), f_0=f always admits a global solution.B. (Existence of closed geodesics). If there exists a compact set KM such that f(S~1)∩K≠φ andE(f)≤(1/π)l(K)~2,then there exists a closed geodesic homotopic to f. If then the closed geodesic is minimal.C. Some estimates about injective radius are obtained.Some example is found showing that the inequalities in B are sharp.     

Minimal length of two intersecting simple closed geodesics

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound L _n on the length attained by the longest of the two geodesics. We show the existence of a surface S _n on which there exists two simple closed geodesics of length L _n intersecting n times and explicitly find L _n for . The first author was supported in part by SNFS grant number 2100-065270, the second author was supported by SNFS grant number PBEL2-106180.     

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.     

Homogeneous geodesics and the critical points of the restricted Finsler function

In this paperwe study the set of homogeneous geodesics of a left-invariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. We extend J. Szenthe’s result on homogeneous geodesics to left-invariant Finsler metrics. This result gives a relation between geodesic vectors and restricted Minkowski norm in Finsler setting. We show that if a compact connected and semisimple Lie group has rank greater than 1, then for every left-invariant Finsler metric there are infinitely many homogeneous geodesics through the identity element.     

Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.