Sergei Yakovenko's blog: on Math and Teaching

Monday, January 30, 2017

Lecture 11 (Jan 16, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 3:54
Tags: , ,

Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane $\mathbb H$, geodesics on it. Realization of the hyperbolic geometry on a surface in $\mathbb R^3_{++-}$. Impossibility of global embedding of $\mathbb H$ into $\mathbb R^3_{+++}$.

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.

Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

1. Pfaffian systems and their integrability
2. From local to global solutions: monodromy
3. Geometric language: covariant derivative and its curvature
4. Meromorphic functions, meromorphic forms
5. Example: multidimensional Euler system
6. Regular singularities
7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

Blog at WordPress.com.