Course Overview

Riemannian Geometry is a basic course for any graduate student in Mathematics who wants to study Geometry, Topology or Dynamic Systems, and is also a relevant course for students of Analysis and Applied Mathematics.

Learning Achievement

Provide to the student the basic tools and some fundamental results of Riemannian Geometry.

Competence

Course prerequisites

Grading Philosophy

Two written tests.

Course schedule

Program: Riemannian metrics; Connections; Completeness; Curvature; Isometric immersions; Variational calculus; Applications. Detailed program: (1) Riemannian metrics; Examples of Riemannian manifolds: the Euclidean space R^n, the sphere S^n, the real hyperbolic space H^n, product of Riemannian manifolds, conformal metrics, Riemannian coverings, flat tori, the Klein bottle, Riemannian submersions, the Hopf fibration and the complex projective space, quotient manifolds, Lie groups. (2) Connections; Parallel transport along a curve; Geodesics; Isometries and Killings vector fields; Induced connections. (3) Completeness; The Hopf-Rinow theorem; Cut locus, Examples. (4) The Riemann-Christoffel curvature tensor; The Ricci tensor and scalar curvature; Covariant derivative of tensors; Examples. (5) Isometric immersions; The second fundamental form; The fundamental equations. (6) Variational calculus; The energy functional; Jacobi vector fields; Conjugate points; Examples. (7) Space forms; The Synge theorem; The Bonnet-Myers theorem; Nonpositively curved manifolds.

Course type

Online Course Requirement

Instructor

Fernando Manfio, Irene Ignazia Onnis

Other information

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