Math 201 Syllabus
week 1:
- quick review of ideas from algebra and topology
- definition and examples of smooth manifolds
- smooth maps between manifolds
- definition of tangent vectors
- the tangent bundle
- vector fields and the Lie bracket
week 2:
- inner products
- Riemannian metrics
- isometries, arc length, and volume
- metrics on Lie groups
- homogeneous spaces
- completeness
Week 3:
- affine connections
- parallel translation
- The Levi-Civita Connection
- the example of the sphere and the hyperbolic plane
Week 4
- definition of geodesics
- existence and uniqueness result
- the exponential map
- minimizing properties of geodesics
- convex neighborhoods