**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Geodesic

Summary

In geometry, a geodesic (ˌdʒiː.əˈdɛsɪk,*-oʊ-,*-ˈdiːsɪk,_-zɪk) is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line".
The noun geodesic and the adjective geodetic come from geodesy, the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.
In a Riemannian manifold or submanifold, geodesics are characterised by the property of hav

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (4)

Related publications (30)

Loading

Loading

Loading

Related units (1)

Related concepts (83)

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.

General relativity

General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current des

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tang

The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.

Related lectures (30)

Related courses (14)

MATH-123(b): Geometry

Ce cours donne une introduction à la géométrie des courbes et des surfaces.

MATH-512: Optimization on manifolds

We develop, analyze and implement numerical algorithms to solve optimization problems of the form: min f(x) where x is a point on a smooth manifold. To this end, we first study differential and Riemannian geometry (with a focus dictated by pragmatic concerns). We also discuss several applications.

PHYS-427: Relativity and cosmology I

Introduce the students to general relativity and its classical tests.

Martin Bauer, Martins Bruveris

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

This thesis is a study of harmonic maps in two different settings. The first part is concerned with harmonic maps from smooth metric measure spaces to Riemannian manifolds. The second part is study of harmonic maps from Riemannian polyhedra to non-positively curved (locally) geodesic spaces in the sense of Alexandrov. The first part is organized as follows. We begin by defining a notion of harmonicity, and justify- ing the definition by checking it against pre-existing definitions and results in special cases. There are two main theorems in this section. The first is Theorem 0.1.1, which is the general- ization of the Shoen-Yau theorem [SY76] in our setting. The second is on the convergence of harmonic maps between Riemannian manifolds. Specifically we will show that if fi : Mi → N are a sequence of harmonic maps between Riemannian manifolds, and if the manifolds Mi converge to a smooth metric measure space M in the measured Gromov-Hausdorff topology, then the fi converge to a harmonic map f : M → N. This is the content of Theorem 0.1.2 In the second part, we prove Liouville-type theorems for harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. To this end we gen- eralize some Liouville- type theorems for subharmonic functions from [Yau76]. Then we study 2-parabolic admissible Riemannian polyhedra and prove vanishing results for subharmonic functions and harmonic maps on 2-parabolic pseudomanifolds.