Syllabus
- Background material on topological spaces
- Open and closed sets,
- Toplogies and topological spaces
- Continuous functions
Supspace and quotient topologiesBasesCompact sets
- Basic concept of a manifold
- Smooth functions and diffeomorphisms
- Coordinate charts, compatibility of charts
- Atlases, equivalence of atlases, maximal atlases
- Hausdorff condition
- Definition of a manifold
- Induced topology, open, closed and compact subsets
- Smooth maps and partitions of unity
- Smooth functions, bump functions, existence
- Smooth maps between manifolds
- Submersions, Immersions, embeddings
- Partitions of unity
- Manifolds are embeddable in euclidean space
- Vectors and vector fields
- Tangent vectors as curves, derivations, and via local coordinates
- Tangent space
- Derivative of a smooth map
Tangent bundleVector fields as smooth sections and derviationsFlow of a vector fieldLie brackets as commutators and via flowsFrobenius Theorem
Differential Forms
Covectors and the Cotangent bundlePullback of a formExterior algebra and wedge productExterior derivativeClosed and exact forms
Integration
Integration of a top degree formIntegration along submanifoldsStokes theorem