This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. Modern differential geometry in its turn strongly contributed to modern physics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Like modern analysis itself, differential geometry originates in classical mechanics. I explain complete integrability of geodesic flows and other Hamiltonian systems after the method developed by Anton Thimm.Differential geometry studies geometrical objects using analytical methods. In a joint lecture with Karsten Grove, we discuss Wiedersehen manifolds, Zoll surfaces, Blaschke manifolds, and harmonic spaces. In these notes I discuss the theorem of Ambrose and Hicks on parallel translation of torsion and curvature and the Lie theoretic description of affine manifolds with parallel torsion and curvature of Nomizu. This work was one of my main motivations for discussing geometric structures. Recall that the latter theorem is crucial in the celebrated work of Benoist, Foulon and Labourie on the regularity of stable foliations of contact Anosov flows. I discuss geometric structures with the main aim of proving a theorem of Singer on the local homogeneity of Riemannian manifolds and Gromov's Open Orbit Theorem. I show that the automorphism groups of certain natural geometric structures are Lie groups with respect to the compact-open topology.Īs a particular application I get that the isometry group of a Riemannian or semi-Riemannian manifold is a Lie group with respect to the compact-open topology. Length and geodesic spaces, length metrics on simplicial complexes, Theorem of Hopf-Rinow for geodesic spaces, upper curvature bounds in the sense of Alexandrov, barycenters, filling discs, cones, tangent cones, spherical joins, Tits buildings, short homotopies, Theorem of Hadamard-Cartan. These notes contain basics on Kähler geometry, cohomology of closed Kähler manifolds, Yau's proof of the Calabi conjecture, Gromov's Kähler hyperbolic spaces, and the Kodaira embedding theorem. (Published by Birkhäuser as DMV Seminar 25, 1995)ĮSI Lectures in Mathematics and Physics. In an appendix, Misha Brin proves Anosov's theorem on the ergodicity of geodesic flows on closed manifolds of negative curvature. I discuss the geometry of metric spaces, spaces of nonpositive curvature, rank one spaces, and rank rigidity. Lectures on Spaces of Nonpositive Curvature.Riemannian distance, theorems of Hopf-Rinow, Bonnet-Myers, Hadamard-Cartan.ĭistance functions and Riccati equation, comparison theory for scalar Riccati equations, Bishop-Gromov inequality and applications, Heintze-Karcher inequalities. In the later version, I also discuss the theorem of Birkhoff / Lusternik-Fet and the Morse index theorem. In these notes, I discuss first and second variation of length and energy and boundary conditions on path spaces. Riemannian immersions and submersions, Gauss and Codazzi equation, O'Neill's formula, projective spaces, Hopf map, Fubini-Study metric. Semi-Riemannian metrics, Levi-Civita connection, curvature. x+169 pp.Ĭonnections on manifolds, geodesics, exponential map.Ī short and elementary exposition of vector bundles and connections on vector bundles. Translated by Walker Stern.Ĭompact Textbooks in Mathematics. Introduction to Geometry and Topology.Einführung in die Geometrie und Topologie.Skript zum ersten Teil meiner Vorlesung für Informatiker.Įrgänzendes Skript zu meinen Vorlesungen über Differentialgeometrie. Lecture Notes Werner Ballmann Kurz und bündig
0 Comments
Leave a Reply. |