Sergei Yakovenko's blog: on Math and Teaching

Wednesday, January 11, 2017

Lecture 10, Jan 9, 2017

Introduction to the Riemannian geometry

  1. Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
  2. Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
  3. Covariant derivative \overline\nabla\text{ on }\mathbb R^n and its properties (symmetry, flatness, compatibility with the scalar product).
  4. Smooth submanifolds of \mathbb R^n. The induced  Riemannian metric and connection. Gauss equation.
  5. Weingarten operator on hypersurfaces and its properties. Gauss map.
  6. Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
  7. Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
  8. Symmetries of the curvature and Ricci tensors.
  9. Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.

The lecture notes are available here.

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