Introduction to the Riemannian geometry
- Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
- Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
- Covariant derivative and its properties (symmetry, flatness, compatibility with the scalar product).
- Smooth submanifolds of . The induced Riemannian metric and connection. Gauss equation.
- Weingarten operator on hypersurfaces and its properties. Gauss map.
- Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
- Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
- Symmetries of the curvature and Ricci tensors.
- Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.
The lecture notes are available here.