**Textbook: Do Carmo, Riemannian
Geometry**

**Grading Policy: There will be two
takehome exams. Homework will be assigned in**
** do
Carmo and checked by the grader but it will not be counted as part
of the final**
** grade.**
**However, it is a fundamental part of developing an understanding of
the**
** subject
and the take-home exams will cover similar problems.**

**Syllabus:**

- Review of Geodesic Flow and Convexity, Do Carmo Chap 3
- Review of Sectional Ricci and Scalar Curvature, Do Carmo Chap 4
- Jacobi Fields and Conjugate Points, Do Carmo Chap 5
- Warped Product Spaces, Handout
- Isometric Immersions and the Second Fundamental Form, Do Carmo Chap 6
- The Gauss, Ricci and Codazzi Equations, Do Carmo Chap 6
- Completeness and the Theorem of Hadamard, Do Carmo Chap 7
- Cartan's Theorem and Spaces of Constant Curvature, Do Carmo Chap 8
- Variations of Energy, Do Carmo Chap 9
- Bonnet-Myers Theorem, Do Carmo Chap 9
- Synge Weinstein Theorem, Do Carmo Chap 9
- Rauch Comparison Theorem, Do Carmo Chap 10
- Index Lemma, Do Carmo Chap 10
- The Morse Index Theorem, Do Carmo Chap 11
- Toponogov Theorem, Cheeger and Ebin Chap 2
- Bishop Volume Comparison Theorem, Bishop and Crittendon
- The Laplacian and Ricci Curvature, Cheeger's Critical Points Paper
- Critical Points of Distance Functions, Cheeger's Crit Pts Paper
- The Grove-Shiohama Sphere Theorem, Cheeger's Crit Pts Paper
- Finiteness Theorems, Cheeger's Crit Points Paper
- Harmonic Functions on Manifolds, Schoen and Yau