Variété riemannienne

Résumé

En mathématiques, et plus précisément en géométrie, la variété riemannienne est l'objet de base étudié en géométrie riemannienne. Il s'agit d'une variété, c'est-à-dire un espace courbe généralisant les courbes (de dimension 1) ou les surfaces (de dimension 2) à une dimension n quelconque, et sur laquelle il est possible d'effectuer des calculs de longueur. En termes techniques, une variété riemannienne est une variété différentielle munie d'une structure supplémentaire appelée métrique riemannienne permettant de calculer le produit scalaire de deux vecteurs tangents à la variété en un même point. Cette métrique permet de définir la longueur d'un chemin entre deux points de la variété, puis les géodésiques qui répondent à un problème de plus court chemin. Les concepts fondamentaux qu'on associe à la variété riemannienne sont la connexion de Levi-Civita et la courbure. Définitions et exemples élémentaires Définition formelle Une variété riemannienne est la donnée

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Differentiable manifolds are a certain class of topological spaces which, in a way we will make precise, locally resemble R^n. We introduce the key concepts of this subject, such as vector fields, differential forms, integration of differential forms etc.

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.

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

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Séances de cours associées (24)

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Let M be a C-2-smooth Riemannian manifold with boundary and N a complete C-2-smooth Riemannian manifold. We show that each stationary p-harmonic mapping u: M -> N, whose image lies in a compact subset of N, is locally C-1,C-alpha for some alpha is an element of (0, 1), provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C-1-smooth weakly p-harmonic mappings from which follows a Liouville-type theorem in the same setting. (C) 2019 Elsevier Ltd. All rights reserved.

2019

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Harmonic Maps on Smooth Metric Measure Spaces and on Riemannian Polyhedra

Zahra Sinaei

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.

EPFL2013

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A Gauss-Bonnet Theorem for Asymptotically Conical Manifolds and Manifolds with Conical Singularities

Adrien Giuliano Marcone

The purpose of this thesis is to provide an intrinsic proof of a Gauss-Bonnet-Chern formula for complete Riemannian manifolds with finitely many conical singularities and asymptotically conical ends. A geometric invariant is associated to the link of both the conical singularities and the asymptotically conical ends and is used to quantify the Gauss-Bonnet defect of such manifolds. This invariant is constructed by contracting powers of a tensor involving the curvature tensor of the link. Moreover this invariant can be written in terms of the total Lipschitz-Killing curvatures of the link. A detailed study of the Lipschitz-Killing curvatures of Riemannian manifolds is presented as well as a complete modern version of Chern's intrinsic proof of the Gauss-Bonnet-Chern Theorem for compact manifolds with boundary.

EPFL2019

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