Related papers: Geometry of geodesics
For an investigation of the physical properties of gravitational fields the observation of massive test particles and light is very useful. The characteristic features of a given space-time may be decoded by studying the complete set of all…
These lecture notes are based on an introductory course given by the author at the summer school "Noncommutative Algebraic Geometry" at MSRI in June 2012. The emphasis throughout is on examples to illustrate the many different facets of…
These notes aim to give an introduction to a few aspects of noncommutative geometry.
These are introductory lecture notes on complex geometry, Calabi-Yau manifolds and toric geometry. We first define basic concepts of complex and Kahler geometry. We then proceed with an analysis of various definitions of Calabi-Yau…
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on…
The article surveys inverse problems related to the twisted geodesic flows on Riemannian manifolds with boundary, focusing on the generalized ray transforms, tensor tomography, and rigidity problems. The twisted geodesic flow generalizes…
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and…
The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples…
In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information…
Symplectic and Poisson geometry emerged as a tool to understand the mathematical structure behind classical mechanics. However, due to its huge development over the past century, it has become an independent field of research in…
This is an expository treatise on the development of the classical geometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the…
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length…
We describe, in an intrinsic way and using the global chart provided by Ito's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These…
Notes prepared for the introductory general relativity course PHYSICS 748 at The University of Auckland. They are designed to introduce general relativity to upper-year undergraduate students directly using the modern language of…
The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…
This entry contains the core material of my habilitation thesis, soon to be officially submitted. It provides a self-contained presentation of the original results in this thesis, in addition to their detailed proofs. The motivation of…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
In this work we discuss the notion of stationary curves of the length functional, the so-called (weak) geodesics, on a Riemannian manifold. The motivation behind this work is to give a detailed description of many key concepts from…
Geodesics deviation equation (GDE) is itroduced. In "adiabatic" approximation exact solution of the GDE if found. Perturbation theory in general case is formulated. Geometrical criterion of local instability which may lead to chaos is…