Riemannian Geometry University of Sao Paulo
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.
Provide to the student the basic tools and some fundamental results of Riemannian Geometry.
Two written tests.
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.
Online Course Requirement
Fernando Manfio, Irene Ignazia Onnis
Site for Inquiry
Please inquire about the courses at the address below.
Email address: http://conteudo.icmc.usp.br/Portal/conteudo/1079/538/foreign-scholars