This course is useful for students of geometry, nonlinear analysis and general relativity.

We study how curvature affects local and global properties of smooth manifolds. The principal tool is to use the curvature behavior of geodesics, which are length minimizing curves. We shall develop the intrinsic, classical and differential form notations in parallel. Later we hope to mention how to generalize to length spaces satisfying synthetic conditions. We shall follow do Carmo's text for the first part of the course. We shall use notes of Shiohama and notes of Petersen for the second part of the course. Topics include (depending on time):

• Riemannian Metrics
• Connections. Affine & Riemannian.
• Geodesics. Minimizing properties of geodesics; convex neighborhoods; geodesic flow.
• Curvature. Sectional curvature; Ricci & scalar curvature; tensors.
• Jacobi fields. Conjugate points.
• Isometric Immersions. Second fundamental form.
• Complete manifolds. Hopf-Rinow Theorem; Hadamard Theorem.
• Space Forms. Cartan's Theorem on recovering the metric from curvature; hyperbolic space and its properties.
• Variation of Geodesics. First and second variation formulae for length and energy of geodesics. Theorems of Bonnet-Meyers-Synge.
• Index Theory of Geodesics. Sturm-Liouville Theory. Rauch Comparison Theorem. Hessian comparison theorem. Laplacian Comparison Theorem and applications.
• Closed Geodesics. Preissman's Theorem.
• Cut locus. Conjugate locus. Injectivity radius.
• Toponogov Theorem. Alexandrov's Theorem. Soul Theorem.
• Sphere Theorem.
• Introduction to Alexandrov Spaces.
• Gromov-Hausdorff convergence.

Last updated: 08 / 17 / 01