Riemannian Geometry II

Textbook:   Do Carmo, Riemannian Geometry
            Cheeger and Ebin, Comparison Theorems in Riemannian Geometry (Chapter 2)
            Bishop and Crittendon,  Geometry of Manifolds (Chapter 11)
            Cheeger, Critical Points of Distance Functions..., in Lecture Notes of Math 1504.
            Schoen and Yau,

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