Graduate course on Differential Geometry, Fall 2023

Description

This is an overview course targeted at all graduate students in mathematics. The goal is to give an introduction to some of the methods and research areas of modern differential geometry and Riemannian geometry.

Prerequisities are preferably some introductory course on differential manifolds (covering the basic notions of smooth and Riemannian manifolds, covariant derivative, geodesics, curvature, for example my course in Spring 2023), and advanced level courses on algebra, analysis, and topology. Lecture notes from my aforementioned master course in differential geometry can be found here.

Lecturer

Klaus Kröncke

Time and place

The lectures takes place Tuesdays 13:15-15:00 (and sometimes Friday, 13:15-15:00) in room 3721 at the math department, starting on Aug 29.

Registration

Via email (kroncke@kth.se), before the course starts.

Content and (preliminary) schedule

Note: References given as numbers refer to the literature below.

I will first give a couple of lectures on global Riemannian geometry, with content described below. I'll mostly follow [5]. Very good sources are also [2] and [7].

• Aug 29: Organization of the lecture, recap of the basics of Riemannian geometry (Covariant derivative, geodesics, curvature). Notes
• Sep 5: Riemannian manifolds as metric spaces, geodesics minimize lengths. Notes
• Sep 12: Complete Riemannian manifolds, Hopf-Rinow theorem. Notes
• Sep 19: Jacobi fields and conjugate points, covering maps and fundamental group. Notes
• Sep 26: Curvature and topology: Cartan-Hadamard and Bonnet-Myers theorems. Notes
• Oct 03: No lecture (conference travel)

Afterwards, we will have some seminar talks on purely geometric comparison results:

• Oct 10: Gauss-Bonnet theorem for surfaces [7, Chapter 9, focus on Theorems 9.3 and 9.7] (or alternatively, [5, Sections 3.6 and 3.7]) Notes
• Oct 13 (Friday!): Classification of manifolds of constant curvature [2, Chapter 8, focus on Sections 2 and 4] Notes
• Oct 31: The Rauch Jacobi field comparison theorem [2, Chapter 10 up to Prop 2.4, if time permits, some comments on the focal point version in Sec 4] Notes
• Nov 7: The Bishop-Gromov volume comparison theorem [4, Section 3H], see also [8, Section 7.1.2] Notes

The remainder of the course consists of seminar talks on the spectral analysis of Laplace operators on manifolds.

• Nov 14: The Laplacian of Riemannian manifolds, function space on manifolds [6, Section 3.1, including relevant definitions and statements from Section A.1 for Section 3.2] Notes
• Nov 21: The spectrum of the Laplacian [6, Section 3.2 until the proof of Theorem 3.2.1]. If time permits, outline briefly how to solve heat and wave equations on manifolds via spectral decomposition [3, Chapter I, Section 4] Notes
• Nov 28: The Laplace operator on differential forms [6, Section 3.3, at least until Corollary 3.3.3] Notes
• Dec 5: No talk (conference travel)
• Dec 12: Cohomology classes and harmonic forms [6, Section 3.4, at least until Corollary 3.4.3] Notes
• Dec 14, room 3418 (Thursday!): The Bochner method and eigenvalue estimates [6, Section 4.5 up to the proof of Theorem 4.5.3]. You might need to explain some concepts in [6, Section 4.3].

Course literature

1. M. Berger, P. Gauduchon, E. Mazet, Le Spectre d'une Variete Riemannienne, Springer Lecture Notes in Mathematics (in french)
2. M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications
3. I. Chavel, Eigenvalues in Riemannian Geometry, Pure and applied mathematics
4. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer Universitext
5. O. Goertsches, Differentialgeometrie, lecture notes (in german)
6.  J. Jost, Riemannian Geometry and Geometric Analysis, Springer Universitext
7. J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer Graduate Texts in Mathematics
8. P. Petersen, Riemannian Geometry, Springer Graduate Texts in Mathematics

Examination

Homework problems and oral presentations.