This course is designed for PhD students in pure or in applied mathematics and should also be of interest to those in mathematical physics.
Smooth manifolds underlie a great deal of modern mathematics: differential geometry, global analysis, the theory of Lie groups, geometric mechanics and much else, as well as large areas of mathematical physics.
The main part of this course will cover the basic theory of smooth manifolds and smooth maps, vector fields and differential forms, and vector bundles. These are irreducible requirements for work with smooth manifolds.
If time permits the course will also cover basic geometry of vector bundles, and introduction to Riemannian geometry.
The course will concentrate on how to work with smooth manifolds. Most proofs will be only sketched, but references for complete arguments will be provided. I hope that at the end of the course you will be able to make use of the literature to learn more of what is particularly important for you in your own work.
There will be twenty live lecture sessions. The pdf slides will be also available in due course.
If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.
Calculus of several variables.
Linear algebra (axioms of a vector space, linear operators in finite dimensions, bases, inner product spaces, dual spaces).
Basic topology of Euclidean spaces (open and closed sets, compactness, open covers).
The differential geometry of curves and surfaces in three dimensional Euclidean space will be used for illustration.
A knowledge of point-set topology is desirable.
Outline syllabus (details will be added gradually):
- Overview of the point-set topology. Topological manifolds. Examples.
- Smooth manifolds, smooth maps, tangent vectors.
- Regular Value Theorem, immersions and embeddings.
- Notion of vector bundle. Basic operations and constructions.
- Vector fields and their flows, Lie derivative.
- Tensor fields and differential forms.
- Connections on vector bundles.
- Topics on Riemannian Geometry (if time permits).