Notes on Differential Geometry and Lie Groups

by Jean Gallier

Introduction to Manifolds and Lie Groups, Review of Groups and Group Actions, Manifolds, Construction of Manifolds From Gluing Data, Lie Groups, Lie Algebra, Exponential Map, The Derivative of exp and Dynkin’s Formula, Bundles, Riemannian Metrics, Homogeneous Spaces, Differential Forms, Integration on Manifolds, Distributions and the Frobenius Theorem, Connections and Curvature in Vector Bundles, Geodesics on Riemannian Manifolds, Curvature in Riemannian Manifolds, Curvatures and Geodesics on Polyhedral Surfaces, The Laplace-Beltrami Operator and Harmonic Forms, Spherical Harmonics, Discrete Laplacians on Polyhedral Surfaces, Metrics and Curvature on Lie Groups, The Log-Euclidean Framework, Statistics on Riemannian Manifolds, Clifford Algebras, Clifford Groups, Pin and Spin, Tensor Algebras.

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