# MA6C0 Differential Geometry

**Lecturers: Shengwen Wang**

**Term(s):** Term 1

**Commitment: **30 lectures

**Assessment:** Oral Exam

**Formal registration prerequisites: **None

**Assumed knowledge:**

MA3H5 Manifolds: Key topics from MA3H5 will be covered rapidly in the first few lectures but you should do this module in Term 2 for a thorough discussion of them

- Bilinear forms
- Eigenvalues and Eigenvectors

- Differentiation of functions of several variables, including the Chain Rule
- Inverse and Implicit Function theorems

MA260 Norms, Metrics and Topologies OR MA222 Metric Spaces:

- Basic point set topology

- Existence and uniqueness of solutions to ODEs and their smooth dependence on parameters and initial conditions

**Useful background:**

**Synergies:**

**Summary:**

- Review of basic notions on smooth manifolds; tensor fields
- Riemannian metrics
- Affine connections; Levi-Civita connection; parallel transport
- Geodesics; exponential map; minimising properties of geodesics
- The curvature tensor; sectional, Ricci and scalar curvatures
- Training in making calculations: switching covariant derivatives; Bochner/WeitzenbĂ¶ck formula
- Jacobi fields; geometric interpretation of curvature; second variation of length
- Classical theorems in Riemannian Geometry: Bonnet-Myers, Hopf-Rinow and Cartan-Hadamard

**Books**:

Lee, J. M.: *Riemannian Manifolds: An Introduction to Curvature*. Graduate Texts in Mathematics, 176. Springer-Verlag, 1997

Gallot, S., Hulin, D., Lafontaine, J.: *Riemannian Geometry*. Springer. 2nd edition, 1993

Jost, J.: *Riemannian Geometry and Geometric Analysis* 5th edition. Springer-Verlag, 2008

Petersen, P.: *Riemannian Geometry Graduate Texts in Mathematics*, 171. Springer-Verlag, 1998

Kobayashi, S., Nomizu, K.: *Foundations of Differential Geometry*

do Carmo, M: *Riemannian Geometry*. BirkhĂ¤user, Boston, MA, 1992