Geodesics on the tangent sphere bundle of a Riemannian manifold (original) (raw)

Geodesics and the geometry of manifolds

2014

In this paper contains some remarks regarding projectively related (metric) connections on a 4-dimensional manifold, that is, Levi- Civita connections which yield the same paths for their (unparametrised) geodesics. Some techniques for such an investigation are brie∞y outlined. The signature of the metric is arbitrary but special emphasis is laid on the Lorentz case and the connection with the Einstein principle of equivalence in general relativity theory. The results are based on joint work with David Lonie and Zhixiang Wang in Aberdeen.

Geodesics in Lorentzian Manifolds

2016

We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between “events”. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called “geodesically complete”, or simply, “complete”. It is easy to show that g(σ′, σ′) is constant if σ is a geodesic and g is a smooth metric on a manifold M , so one can characterize geodesics in terms of their causal character : if g(σ′, σ′) 0, σ is spacelike. If g(σ′, σ′) = 0, then σ is lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the not...

Powers of the space forms curvature operator and geodesics of the tangent bundle

It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space form and give a unified proof of the following property: all geodesic curvatures of projected curve are zero starting from k_3,k_6 and k_{10} for the real, complex and quaternionic space formes respectively.

Geodesics and near-geodesics in the manifolds of projector frames

Linear Algebra and its Applications, 1988

The set of projector frames (i.e. finite sequences of commuting projectors which sum to the identity) can be given a differentiable manifold structure with an affine connection. We compare the resulting geodesic arcs with other naturally arising paths and find them cubically close. We also discuss Riemannian geodesics in an appropriate Hilbert-space context.

Geodesic Vector Fields on a Riemannian Manifold

Mathematics

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

The curvature: a variational approach

The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot-Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, we extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces. Contents Chapter 1. Introduction v 1.1. Structure of the paper viii 1.2. Statements of the main theorems viii 1.3. The Heisenberg group x Part 1. Statements of the results Chapter 2. General setting 2.1. Affine control systems 2.2. End-point map 2.3. Lagrange multipliers rule 2.4. Pontryagin Maximum Principle 2.5. Regularity of the value function Chapter 3. Flag and growth vector of an admissible curve 3.1. Growth vector of an admissible curve 3.2. Linearised control system and growth vector 3.3. State-feedback invariance of the flag of an admissible curve 3.4. An alternative definition Chapter 4. Geodesic cost and its asymptotics 4.1. Motivation: a Riemannian interlude 4.2. Geodesic cost 4.3. Hamiltonian inner product 4.4. Asymptotics of the geodesic cost function and curvature 4.5. Examples Chapter 5. Sub-Riemannian geometry 5.1. Basic definitions 5.2. Existence of ample geodesics 5.3. Reparametrization and homogeneity of the curvature operator 5.4. Asymptotics of the sub-Laplacian of the geodesic cost 5.5. Equiregular distributions 5.6. Geodesic dimension and sub-Riemannian homotheties 5.7. Heisenberg group 5.8. On the "meaning" of constant curvature Part 2. Technical tools and proofs Chapter 6. Jacobi curves 6.1. Curves in the Lagrange Grassmannian 6.2. The Jacobi curve and the second differential of the geodesic cost 6.3. The Jacobi curve and the Hamiltonian inner product 6.4. Proof of Theorem A 6.5. Proof of Theorem D Chapter 7. Asymptotics of the Jacobi curve: equiregular case 7.1. The canonical frame 7.2. Main result iii 7.3. Proof of Theorem 7.4 7.4. Proof of Theorem B 7.5. A worked out example: 3D contact sub-Riemannian structures Chapter 8. Sub-Laplacian and Jacobi curves 8.1. Coordinate lift of a local frame 8.2. Sub-Laplacian of the geodesic cost 8.3. Proof of Theorem C

On the problem of geodesic mappings and deformations of generalized Riemannian spaces

Journal of Mathematical Analysis and Applications, 2017

This paper is devoted to a study of geodesic mappings and infinitesimal geodesic deformations of generalized Riemannian spaces. While a geodesic mapping between two generalized Riemannian spaces any geodesic line of one space sends to a geodesic line of the other space, under an infinitesimal geodesic deformation any geodesic line is mapped to a curve approximating a geodesic with a given precision. Basic equations of the theory of geodesic mappings in the case of generalized Riemannian spaces are obtained in this paper. A new generalization of the famous Levi Civita's equation is found. Necessary and sufficient conditions for a nontrivial infinitesimal geodesic deformation are given. It is proven that a generalized Riemannian space admits nontrivial infinitesimal geodesic deformations if and only if it admits nontrivial geodesic mappings. At last it is shown that generalized equidistant spaces of primary type admit nontrivial geodesic deformations.