Syllabus
Basic notions from general topology. Topological and differentiable manifolds, maps between manifolds.
Submanifolds in the Euclidean space. Tangent spaces, tangent maps, vector fields, Lie bracket of vector fields.
Affine connection on a manifold as differentiation of vector fields. The Levi-Civita connection on a manifold in R^n.
The parallel transport along curves, geodesic curves - definitions and existence theorems. Exponential map at a point. The torsion tensor field and the curvature tensor field, its geometric meaning.
Riemannian (pseudo-Riemannian) metric, the induced structure of a metric space. The Riemannian connection - existence and uniqueness, relationship with the Levi-Civita connection (on a submanifold with induced metric).
The Gaussian formula and its geometric interpretation for surfaces - Gauss theorem. The Gauss curvature of a surface.
The sectional curvature of a Riemannian manifold, spaces with constant curvature. Extremal properties of geodesics.
Global properties of geodesics on a complete Riemannian manifold.
Annotation
The goal of this lecture is to acquaint the students with one of the basic techniques of the Mathematical Physics, namely pseudo-Riemannian geometry.