We will follow the book "An Introduction to Manifolds" by Loring W. Tu. (Second Edition).
§1-9, 11.1-11.4, 12.1, 12.2, §13-16, 17.1-17.6, 18.1-18.7, §19, 21.1-21.4, §22, 23.1-23.6, §24, 25.1-25.3, 26.1-26.23, 27.1-27.4, §29, Appendices B.3, §C.
Through the university's web system, you can access this book here: https://link.springer.com/book/10.1007%2F978-1-4419-7400-6
The following is a list of the most central themes that are relevant for discussion during the oral exam.
- Smooth manifolds
- Smooth maps
- Partitions of unity
- Tangent spaces and differentials
- Submanifolds
- Quotient spaces
- Immersions and submersions
- Tangent bundles
- Vector fields and Lie brackets
- Flows of vector fields
- Lie groups
- Co-tangent spaces/bundles and differential 1-forms
- Tensors and alternating tensors
- Wedge product and exterior algebra
- Differential forms
- Exterior derivative
- Orientations
- Stoke’s theorem
- de Rham cohomology
- Mayer-Vietoris sequence