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
- Integration on manifolds
- Stoke’s theorem
- de Rham cohomology
- Mayer-Vietoris sequence
- Homotopy invariance