Syllabus
1. Topological manifold (charts, transition functions, atlas), smooth manifolds (differential structure), basic examples of manifolds.
2. Smooth maps between manifolds, smooth functions, diffeomorphisms; tangent vectors in a point, tangent space in a point, coordinates on tangent space, geometrical interpretation of vectors; tangent map to a smooth map, coordinate description, Jacobians.
3. A summary of properties of tensor algebra of a vector space; outer algebra of a vector space, basic properties of outer multiplication; symmetric algerba of a vector space, orientation of a vector space, volume of a paralleliped using outer product and the Gramm matrix.
4. Tensor fields on a manifold, Riemann (pseudo)-metric on a manifold, Minkowski spacetime, algebra of differnetial forms as a modul over the ring of functions, orientation of a manifold; de Rham differential in coordinates and without coordinates, exact and closed forms, de Rham complex, de Rham cohomology, Poincare lemma; inverse image of tensor fields and forms by a smooth map, coordinate description, basic properties.
5. Manifolds with a boundary, its tangent space, differential forms on manifolds with boundary, orientation.
6. Integration of forms on a manifold with boundary, Stokes theorem.
7. Volume form on a (pseudo)-Riemannian manifold, integration of functions on such manifolds, local computations. If possible:
8. Lie derivative of vector fields, contraction of forms by vector fileds, connection with the de Rham differential.
Annotation
One of the basic courses in the area of general differential geometry. A recommended course for specialization
Mathematical Structures within General Mathematics.