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.      

Small volume closed hyperbolic 4-manifolds

Further examples of non-orientable compact hyperbolic 4-manifoldsof volume 32²/3 arising from torsion-free subgroups of the [5,3, 3, 3] Coxeter group are given. These are the smallest knownclosed hyperbolic 4-manifolds and arise by consideration ofmaps from the [5, 3, 3, 3] Coxeter group onto the simple simplecticgroup S₄(4).     

On maximal globally hyperbolic vacuum space-times

We prove existence and uniqueness of maximal global hyperbolic developments of vacuum general relativistic initial data sets with initial data (g, K) in Sobolev spaces ${H^{s} \bigoplus H^{s - 1}, \mathbb{N} \ni s > n/2 + 1}$ .     

Torsion and closed geodesics on complex hyperbolic manifolds

Summary LetX be a compact complex manifold covered by complex hyperbolicn-space with the induced metric. Each stable horocycle has a cocomplex structure preserved by the geodesic flow. To a closed geodesic one can thus associate a piece of the Poincaré map with a holomorphic fixed point. The resulting Atiyah-Bott fixed point indices, together with the length and multiplicity of as a periodic orbit, determine the contribution of to certain zeta functionsR _p(z), 0pn. From the leading coefficient ofR _p atZ=0 and the Hodge numbersh ^ij (X) we calculate the Ray-Singer -torsionT _p (X). This indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category.Corresponding results hold for the -torsion of a flat unitary bundle, extending certain formulas of Ray and Singer to the casen>1.Partially supported by the Sloan Foundation and the National Science Foundation     

Minimum volume hyperbolic 3-manifolds

We enumerate the small-volume manifolds that can be obtainedby Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai,Meyerhoff, and the author. In so doing we complete the proofthat the Weeks manifold is the compact hyperbolic 3-manifoldof minimum volume, as well as enumerating the ten smallest one-cuspedhyperbolic 3-manifolds. Received October 21, 2008.     

Cusp transitivity in hyperbolic 3-manifolds

Geometriae Dedicata - In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove...     

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     

Length multiplicities of hyperbolic 3-manifolds

LetM = ℍ³/Γ be a hyperbolic 3-manifold, where Γ is a non-elementary Kleinian group. It is shown that the length spectrum ofM is of unbounded multiplicity.     

A new class of compact spacetimes without closed causal geodesics

We construct the first examples of geodesically complete compact spacetimes admitting a regular globally hyperbolic covering, but which do not contain closed causal geodesics.      

Isometry transformations of hyperbolic 3-manifolds

The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold.     

On lengths of filling closed geodesics of hyperbolic punctured Riemann spheres

Let S be a Riemann sphere with n ≥ 4 points deleted. In this article we investigate certain filling closed geodesics of S and give quantitative common lower bounds for the hyperbolic lengths of those geodesics with respect to any hyperbolic structure on S (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.