Syllabus
Smooth manifolds, differential forms. Distributions, Frobenius theorem (2 versions).
Fibre bundles, transition functions, vector fibre bundles, their local description, classifying maps.
Covariant derivatives on vector bundles, parallel transport, curvature, structure equations.
Homogeneous spaces, the Maurer--Cartan form, the Darboux derivative, fundamental theorem of calculus.
Principal fibre bundles, associated fiber bundles, principal connections and their curvature, structure equations.
Holonomy and monodromy. Calibration (Yang-Mills) fields.
Annotation
The course is a continuation of the course 'Introduction to analysis on manifolds'.
It is a basic course needed for a further study of differential geometry and global analysis, as well as for applications in mathematical physics (Yang-Mills fields).