David Lindemann: Differential Geometry

In the lectures we will cover the basic concepts of modern differential geometry. Differential geometry studies smooth manifolds, that is, geometric objects that, roughly speaking, locally look like ℝⁿ (and whose global topological properties are not too weird, see for example the so-called ‘long line’). You already know examples such as the n-sphere Sⁿ, or smooth surfaces in ℝ³ from analysis. On smooth manifolds we will study a number of constructions and structures, such as vector fields, metrics and various curvature concepts. In addition, smooth manifolds are suitable as spaces for ordinary and partial differential equations, which allow different global topological properties compared to regions in ℝⁿ (e.g. PDEs on the Klein bottle or on the real-projective spaces ℝPⁿ). A focus of this lecture will be submanifolds and induced geometric structures. We will also look at some topics from the perspective of the calculus of variations, e.g. geodesics as critical points of the energy functional. This lecture is also expressly suitable for students of physics courses, as differential geometry represents a fundamental theoretical basis for many modern theories in physics (especially ART, gauge theories such as Yang-Mills -Theory, SuSy, SuGra,…).

1. Smooth manifolds
2. Smooth maps and the IFT
3. Tangent spaces 1
4. Tangent spaces 2
5. Submanifolds
6. Vector bundles
7. The tangent bundle
8. Lie bracket of vector fields, integral curves, flows
9. Infinitesimal generators of one-parameter groups of diffeomorphisms and the Lie derivative of vector fields
10. Dual bundles, 1-forms, and the Whitney sum
11. Tensor bundles and tensor fields
12. Pseudo-Riemannian manifolds
13. Traces, raising and lowering indices, and vector bundles along pseudo-Riemannian submanifolds
14. Local frames and Killing vector fields
15. Connections in vector bundles
16. Parallel transport and the Levi-Civita connection
17. Geodesics and the exponential map
18. Curvature
19. Geodesics and curvature of psuedo-Riemannian submanifolds

These videos were produced by David Lindemann for Universität Hamburg. More information about this course can be found here.