On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.     

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

On Manifolds Admitting Continuous Foliations by Geodesics

We show that complete, simply connected Riemannian manifolds admitting continuous foliations by geodesics with integrable orthogonal distributions are homeomorphic to products F×R. Moreover, the geodesics in the foliation are global minimizers.     

A Classification of Locally Homogeneous Affine Connections with Skew-Symmetric Ricci Tensor on 2-Dimensional Manifolds

 We classify, in an explicit form, the locally homogeneous torsionless affine connections as in the title. We also give some motivation for this research coming from the study of Osserman spaces. (Received 22 July 1999; in revised form 13 January 2000)     

Complete Flat Affine and Lorentzian Manifolds

Following in the tradition of Hilbert's 18th problem of classifying crystallographic groups, we provide a survey of a series of results which have culminated in the study of flat Lorentz manifolds. In particular, Milnor asked whether all complete flat affine manifolds have virtually polycyclic fundamental groups. Margulis answered this question negatively by constructing complete flat Lorentz manifolds with free fundamental groups. In this paper, we follow the effort to classify and understand these interesting counterexamples to Milnor's question, and their generalizations.     

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their sets of homogeneous geodesics through a point. Dedicated to the memory of Professor Aldo Cossu Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce.     

Geometric Integration Algorithms on Homogeneous Manifolds

Abstract. Given an ordinary differential equation on a homogeneous manifold, one can construct a ``geometric integrator' by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is ``full,' then the order of accuracy of orbit capture (i.e., approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere.     

两个非紧致齐性复解析流形

本文给出两个非紧致的齐性复解析流形．用它的齐性子流形构造出两个例外对称典型域的扩充空间，并由复流形上的运动群在超圆上的限制得到了两个例外对称典型域的仿射自同胚群，它们是闭的辛子群．     

Three-Dimensional Naturally Reductive Homogeneous Lorentzian Manifolds

We give a complete classification of three-dimensional naturally reductive Lorentzian spaces, using a classification of the self adjoint Ricci operators.      

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L²_tH^1/2_x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?dinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L ¹ → L ^∞ dispersive estimates still hold without loss for e ^itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.     

A Class of Homogeneous Einstein Manifolds

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c·g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.     

Nonvertical Affine Manifolds and Separation Theorems in Product Spaces

In the paper we introduce a general separation concept for sets in product spaces X × Z, where Z is partially ordered. Using some results about nonvertical affine manifolds we formulate some concerte separation theorems. Especially, we get assertions about the separation of sets by affine mappings, in which the order relation “≦” is replaced by the relation ??”?. Thus, these assertions are suitable to characterize PARETO optimal solutions of vector optimization problems.     

On Affine Normal Forms and a Classification of Homogeneous Surfaces in Affine Three-Space

We classify homogeneous surfaces in real and complex affine three-space. This is achieved by choosing affine coordinates so that the surface is defined by a function whose Taylor series is in a preferred normal form.     

Affine Biharmonic Curves in 3-Dimensional Homogeneous Geometries

Every 3-dimensional Riemannian manifold with 4-dimensional isometry group admits a normal almost contact structure compatible to the metric. In this paper we study affine biharmonic curves in model spaces of Thurston geometry except Sol.     

Closed-form Geodesics and Optimization for Riemannian Logarithms of Stiefel and Flag Manifolds

Journal of Optimization Theory and Applications - We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by Hüper, Markina and Silva...     

Behavior of Distant Maximal Geodesics in Finitely Connected Complete Two-Dimensional Riemannian Manifolds II

We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ² the Cohn–Vossen theorem states that the total curvature of M, say c(M), is 2. We already studied the case c(M)<2 in our previous paper. So we study the behavior of geodesics in M with total curvature 2 in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.