Extensive notes for a one-semester graduate course on differential geometry, written by Clayton Shonkwiler.

The goal was to cover a number of standard topics (vector fields, differential forms and integration, Lie groups and homogeneous spaces, and Riemannian geometry) with an emphasis on symmetry and on simple examples, especially examples related to frame theory.

The notes are available either as a PDF or as an accessible web page.

Here is the table of contents:

1. Manifolds and Vector Fields

   1. Manifolds and Maps
   2. Tangent Vectors
   3. The Differential
   4. Immersions and Embeddings
   5. Vector Fields
   6. The Lie Bracket on Vector Fields
   7. Integral Curves and Lie Derivatives
   8. Extended Example on S³

2. Differential Forms

   1. Integration, Forms, and an Informal Definition
   2. Differential Forms and Vector Calculus
   3. Tensor Algebras and Tensor Fields
   4. Exterior Algebras and Differential Forms
   5. The Exterior Derivative
   6. Pullbacks
   7. Lie Derivatives and Cartan's Magic Formula
   8. De Rham Cohomology
   9. Integration on Manifolds
   10. Integration on Submanifolds
   11. Stokes' Theorem

3. Lie Groups and Homogeneous Spaces

   1. Some Motivation from de Rham Cohomology
   2. Lie Groups
   3. Lie Algebras
   4. Matrix Groups
   5. Adjoint and Coadjoint Representations
   6. Lie Algebra Homomorphisms
   7. Distributions and the Frobenius Theorem
   8. Distributions and Differential Ideals
   9. Lie Group Homomorphisms are Determined by their Differentials
   10. The Exponential Map
   11. Homogeneous Manifolds
   12. Stiefel and Grassmann Manifolds

4. Riemannian Manifolds

   1. Riemannian Metrics
   2. Affine Connections
   3. The Levi-Civita Connection
   4. Geodesics
   5. Geodesics on Lie Groups
   6. Geodesics on Grassmannians
   7. The Curvature Tensor
   8. Sectional Curvature
   9. Ricci and Scalar Curvatures
   10. Gradients and Optimization