Riemannian Geometry

ABOUT THE COURSE:It will be a rigorous course on Riemannian geometry. The course will begin with a brief review of smooth manifolds and the geometry of curves and surfaces. I will then introduce Riemannian metrics and cover some standard topics such as Levi-Civita connection and associated curvature, geodesics, completeness, 1st and 2nd variation formulae and Jacobi fields. If time permits, the course will end with more advance topics, such as comparison theorems (Myers, Bishop-Gromov etc.) and the Bochner technique.A justification for the course: The course aims to fill a much needed gap in the differential geometry courses offered in NPTEL and will equip the students with the necessary background to branch out into various advanced topics in differential geometry and geometric analysis. At some point in the future, the instructor also plans to offer a course in geometric analysis on NPTEL, and this would be a pre-requisite for that.INTENDED AUDIENCE: Masters and PhD students, advanced undergraduatesPREREQUISITES: Multivariable calculus and Smooth manifolds is a must (so must be familiar with calculus on manifolds, tensors, forms and Stokes’ theorem). Helpful to have had a course on curves and surfaces, but my course would be independent of that.

Syllabus

Week 1: Introduction, review of curves and surfaces, Review of smooth manifolds – tensors, differential forms, Stokes’ theorem
Week 2:Introduction to Riemannian metrics, examples and basic constructions
Week 3:Levi-Civita connection, induced connection on tensors and forms, Parallel transport
Week 4:Curvature of Levi-Civita connection, Sectional curvature, Ricci curvature and scalar curvature, geometry of sub-manifolds
Week 5:Geodesics, first variation formula
Week 6:Local behaviour of geodesics, exponential map, normal coordinates
Week 7:Metric geometry, Hopf-Rinow, regularity of distance function
Week 8:2nd variation formula, Jacobi fields, index form
Week 9:Jacobi fields (cont.), characterization of space forms
Week 10:Catch-up (if on schedule, then move onto comparison geometry)
Week 11:Overview of comparison geometry (Rauch, Myer and Bishop-Gromov)
Week 12:Bochner technique