Theoretical / proof-based
Differential geometry · Graduate · Math
Learning objectives from the Differential geometry syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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
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)
Definitions and structure
- 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
Proofs and reasoning
- Differential forms — Differential forms
- Exterior derivative — Exterior derivative
- Integration on manifolds — Integration on manifolds
- Stokes' theorem — Stokes' theorem
- Riemannian metrics — Riemannian metrics
Abstraction and generalization
- 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
Modeling and computation
- Apply smooth manifolds, charts, and atlases in engineering contexts — Apply smooth manifolds, charts, and atlases in engineering contexts
- Apply smooth maps and diffeomorphisms in engineering contexts — Apply smooth maps and diffeomorphisms in engineering contexts
- Apply submanifolds in engineering contexts — Apply submanifolds in engineering contexts
- Apply tangent vectors and the tangent bundle in engineering contexts — Apply tangent vectors and the tangent bundle in engineering contexts
- Apply vector fields in engineering contexts — Apply vector fields in engineering contexts
Data and technology
- Use software to explore differential geometry problems numerically — Use software to explore differential geometry problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Differential forms — Differential forms
- Exterior derivative — Exterior derivative
- Integration on manifolds — Integration on manifolds
- Stokes' theorem — Stokes' theorem
- Riemannian metrics — Riemannian metrics
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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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.