Differential geometry
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Differential Geometry — scope drawn from open differential geometry notes and typical US graduate differential geometry syllabi.
Manifolds and tangent bundles
- Smooth manifolds, charts, and atlases — Smooth manifolds, charts, and atlases
- Smooth maps and diffeomorphisms — Smooth maps and diffeomorphisms
- Submanifolds — Submanifolds
- Tangent vectors and the tangent bundle — Tangent vectors and the tangent bundle
- Vector fields — Vector fields
- Differential forms — Differential forms
- Exterior derivative — Exterior derivative
- Integration on manifolds — Integration on manifolds
- Stokes' theorem — Stokes' theorem
Riemannian geometry
- Riemannian metrics — Riemannian metrics
- Levi-Civita connection — Levi-Civita connection
- Geodesics and the exponential map — Geodesics and the exponential map
- Gauss lemma — Gauss lemma
- Curvature tensor — Curvature tensor
- Ricci curvature — Ricci curvature
- Gauss–Bonnet theorem (introduction) — Gauss–Bonnet theorem (introduction)
- Comparison geometry (overview) — Comparison geometry (overview)
Lie groups and fiber bundles
- Lie groups and Lie algebras — Lie groups and Lie algebras
- Left-invariant vector fields — Left-invariant vector fields
- Exponential map — Exponential map
- Principal — Principal
- Vector bundles (introduction) — Vector bundles (introduction)
- Connections and parallel transport — Connections and parallel transport
- Characteristic classes: Chern classes (introduction) — Characteristic classes: Chern classes (introduction)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Manifolds and tangent bundles
Smooth manifolds, charts, and atlases
Coming soon - Riemannian geometry
Riemannian metrics
Coming soon - Lie groups and fiber bundles
Lie groups and Lie algebras
Coming soon
Notes
Graduate Differential Geometry — scope drawn from open differential geometry notes and typical US graduate differential geometry syllabi. Topic outline: `content/topics/graduate/diff_geometry.json`.