Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives

Given a geometrically finite hyperbolic cone-manifold, with the cone-singularity sufficiently short, we construct a one-parameter family of cone-manifolds decreasing the cone-angle to zero. We also control the geometry of this one-parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.

     

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. In memory of Robert Brooks     

Hyperbolic metric,curvature of geodesics and hyperbolic discs in hyperbolic plane domains

The purpose of this paper is to investigate the relations among some geometric quantities defined for every hyperbolic plane domain of any connectivity, each of which measures, in some sense, how much the domain deviates either from a disc, convex domain, or simply connected domain on one hand, or a punctured domain on the other hand. Supported by the Landau Center for Mathematical Research in Analysis.     

Non-closed isometry-invariant geodesics

Let c be a non-closed and bounded geodesic in a complete Riemannian manifold M. Assume that c is invariant under an isometry A of M and that c is not contained in the set of fixed points of A. We prove that, for some \({k\ge 2}\), the geodesic flow line ? corresponding to c is dense in a k-dimensional torus N embedded in TM. In particular, every geodesic with initial vector in N is A-invariant.     

The space of geodesics

Let M be a manifold with linear connection . The space G(M) of all geodesics of M may be given a topological structure and may be realized as a quotient space of the reduced tangent bundle of M. The space G(M) is a T ₁ space iff the image of each geodesic is a closed subset of M. It is Hausdorff iff each tangentially convergent sequence of geodesics converges in the Hausdorff limit sense to the limit geodesic. If M has no conjugate points and G(M) is Hausdorff, then M is geodesically connected.Supported in part by NSF grant DMS-8803511.     

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.     

Closed geodesics on orbifolds

In this paper, we try to generalize to the case of compact Riemannian orbifolds Q some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds M. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics.In the classical case the solution of those problems involve the consideration of the homotopy groups of M and the homology properties of the free loop space on M (Morse theory). Those notions have their analogue in the case of orbifolds. The main part of this paper will be to recall those notions and to show how the classical techniques can be adapted to the case of orbifolds.     

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...     

Constrained optimization along geodesics

An arc method is presented for solving the equality constrained nonlinear programming problem. The curvilinear search path used at each iteration of the algorithm is a second-order approximation to the geodesic of the constraint surface which emanates from the current feasible point and has the same initial heading as the projected negative gradient at that point. When the constraints are linear, or when the step length is sufficiently small, the algorithm reduces to Rosen's Gradient Projection Method.     

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.     

Polyhedra with simple dense geodesics

We characterize polyhedral surfaces admitting a simple dense geodesic ray and convex polyhedral surfaces with a simple geodesic ray.     

Peacock geodesics in Wasserstein space

Martingale optimal transport has attracted much attention due to its application in pricing and hedging in mathematical finance. The essential notion which makes martingale optimal transport different from optimal transport is peacock. A peacock is a sequence of measures which satisfies convex order property. In this paper we study peacock geodesics in Wasserstain space, and we hope this paper can provide some geometrical points of view to look at martingale optimal transport.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

Quasihyperbolic geodesics in convex domains

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesies are quasiconvex in the norm metric in convex domains in all normed spaces.