SF2722 Differential Geometry

Course at advanced level (course number KTH: SF2722, SU: MM8022), 7.5 credits, spring 2021.

ZOOM room for lectures: https://kth-se.zoom.us/j/67839417630 Links to an external site.
ZOOM room for exercise sessions: https://kth-se.zoom.us/j/69019331136 Links to an external site.

Discussion forum: We encourage everyone to participate actively in the discussion forum.

Teachers: Mattias Dahl and Hans Ringström.

Exercise sessions: Bernardo Fernandes

Administration: Contact the student affairs office for registration and other administrative matters.

Schedule: see here Links to an external site.. 

Literature: 

• Christian Bär, lecture notes Differential geometry Links to an external site..
• John M. Lee, Introduction to smooth manifolds Links to an external site. (KTH library link).
• Peter Petersen, Riemannian Geometry Links to an external site. (KTH library link).
• Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian Geometry Links to an external site. (KTH library link).

As additional reading we recommend 

• John M. Lee, Introduction to topological manifolds;
• John M. Lee, Riemannian manifolds: an introduction to curvature;
• Jeffrey M. Lee, Manifolds and differential geometry;
• William M. Boothby, An introduction to differentiable manifolds and riemannian geometry;
• Barrett O'Neill, Semi-riemannian geometry;
• Ben Andrews, Lectures on differential geometry Links to an external site.;
• Christian Bär, lecture notes Theory of relativity Links to an external site..

Description: The central objects in modern differential geometry are differentiable manifolds. In this course we will study differentiable manifolds and see how they are used to define concepts from analysis in a coordinate-independent way. In an introduction to (semi-)Riemannian geometry we will see how curvature is described. Ideas and methods from differential geometry are fundamental in modern physical theories.

Prerequisites: SF1677 Foundations of Analysis, or corresponding background. Also recommended is SF1678 Groups and Rings, or equivalent. A good knowledge of calculus of several variables including the inverse and implicit function theorems is an important prerequisite.

Content: The course is divided into seven modules with the following content:

• Module 1: Manifolds. This module covers the basic notions of a manifold,  tangent vectors and vector fields. It corresponds to chapter 1 in Bär's notes.
• Module 2: Semi-Riemannian Geometry. In this module, the notion of a Semi-Riemannian metric is introduced, as well as the notion of parallel transport and geodesics. It corresponds to chapter 2 in Bär's notes.
• Module 3: Curvature. This module covers several notions of curvature: the Riemann curvature tensor, sectional curvature, Ricci curvature and scalar curvature. Jacobi fields are also discussed. It corresponds to chapter 3 in Bär's notes.
• Module 4: Submanifolds. In this module, we discuss how the geometry induced on a submanifold relates to the geometry of the ambient space. It corresponds to chapter 4 in Bär's notes.
• Module 5: Flows and Lie brackets. This module covers integral curves and flows of vector fields, as well as the notion of a Lie bracket and a simple version of Frobenius theorem. It corresponds to a part of chapters 8, 9 and 19 in Lee's book. We will also give an introduction to Killing vector fields, following chapter 8 in the book by Petersen.
• Module 6: Riemannian geometry. In this module, we give examples of how local geometric conditions can influence the global topology of a manifold. It corresponds to chapter 5 of Bär's notes.
• Module 7: Continuation of Riemannian geometry. Differential forms, Stokes' theorem and de Rham cohomology. In this module we first continue the study of Riemannian geometry, following parts of chapter 3 in the book by Gallot-Hulin-Lafontaine. After that we look at differential forms, integration thereof, Stokes' theorem and de Rham cohomology. This corresponds to parts of chapters 14-18 in Lee's book.