- Manifolds, smooth maps. Examples. Smooth functions.
- Bundles, vector bundles. Examples: trivial and nontrivial bundles.
- Vector fields on a manifolds as linear operators.
- Differential forms, external derivation.
- Connections on sections of vector bundles.
- Examples: connection on a trivial vector bundle. Linear connection.
- Connection on a vector bundle defined through a projector.
- Curvature of a connection. Difference of connections.
- Characteristic classes. Chern class and Chern character of a connection.
- Magnetic monopole as a connection of a nontrivial line bundle over the sphere.
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