A self-contained, quick and "to the point" cheat sheet of the important definitions and theorems found within differential geometry, for use in applications such as general relativity, information geometry, etc. Only requires basic knowledge of linear algebra, discrete math and some topology, with relevant topics reviewed along the way.
When I started learning differential geometry, I quickly found myself wanting a clear, concise reference manual for the important, and often hierarchical definitions found within the literature. This is my attempt at making one. Primarily adapted from Spivak's A Comprehensive Introduction to Differential Geometry, 3rd Edition, Mendelson's Introduction to Topology, 3rd Edition and Michor’s Topics in Differential Geometry.
Topics in bold are finished (besides possible minor future tweaks), while topics in italics will be added in future updates.
- Prerequisites (some topology, linear algebra and analysis/discrete math)
- Manifolds (differential manifolds and maps between them)
- The Tangent Bundle (tangent spaces and bundles, and their connection to derivatives and vector fields)
- The Cotangent Bundle (dual and cotangent bundles; differentials; contravariant and covariant vector fields)
- Tensors, Part 1 (multilinear functions; covariant tensor bundles; covariant tensor fields and their properties)
- Tensors, Part 2 (contravariant tensor fields and their properties; mixed tensor fields and their properties; contractions)
- Riemannian Manifolds