Notes on Differential Geometry
by Matt Visser
Publisher: Victoria University of Wellington 2011
Number of pages: 246
In this text the author presents an overview of differential geometry, also known as the theory of manifolds. Topics covered: Topological Manifolds and differentiable structure; Tangent and cotangent spaces; Fibre bundles; Geodesics and connexions; Riemann curvature; Exterior differential forms; Lie derivatives; etc.
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by Nigel Hitchin
The historical driving force of the theory of manifolds was General Relativity, where the manifold is four-dimensional spacetime, wormholes and all. This text is occupied with the theory of differential forms and the exterior derivative.
by Theodore Shifrin - University of Georgia
Contents: Curves (Examples, Arclength Parametrization, Frenet Frame); Surfaces: Local Theory (Parametrized Surfaces, Gauss Map, Covariant Differentiation, Parallel Translation, Geodesics); Surfaces: Further Topics (Holonomy, Hyperbolic Geometry,...).
by Dmitri Zaitsev - Trinity College Dublin
From the table of contents: Chapter 1. Introduction to Smooth Manifolds; Chapter 2. Basic results from Differential Topology; Chapter 3. Tangent spaces and tensor calculus; Tensors and differential forms; Chapter 4. Riemannian geometry.
by John Edward Campbell - Clarendon Press
Contents: Tensor theory; The ground form when n=2; Geodesics in two-way space; Two-way space as a locus in Euclidean space; Deformation of a surface and congruences; Curves in Euclidean space and on a surface; The ruled surface; Minimal surface; etc.