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