Differential geometry
Graduate · Math
Syllabus focus
Theoretical / proof-based
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$1,162 · Differential geometry · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle or Venmo.
Topics typically covered
Theoretical / proof-based
Manifolds and tangent bundles
- Smooth manifolds, charts, and atlases
- Smooth maps, diffeomorphisms, and submanifolds
- Tangent vectors, tangent bundle, and vector fields
- Differential forms and exterior derivative
- Integration on manifolds and Stokes' theorem
Riemannian geometry
- Riemannian metrics and Levi-Civita connection
- Geodesics, exponential map, and Gauss lemma
- Curvature tensor; sectional and Ricci curvature
- Gauss–Bonnet theorem (introduction)
- Comparison geometry (overview)
Lie groups and fiber bundles
- Lie groups and Lie algebras
- Left-invariant vector fields and exponential map
- Principal and vector bundles (introduction)
- Connections and parallel transport
- Characteristic classes: Chern classes (introduction)
Notes
Topics reflect common graduate differential geometry syllabi at US universities. First courses often focus on smooth manifolds and Riemannian basics; advanced topics vary by instructor.