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Riemannian geometry Description

Key Elements

Code

MATH 406

Formation

M1 Mathematics

Semester

2

Credits

6

Number of Teaching Hours

36

Number of Tutoring Sessions

24

Number of Laboratory Sessions

0

This course is optional

Content

Objective

Content

• Riemannian metrics • Linear connection: Definition, geometrical interpretation, covariant derivation, parallel transport, torsion and curvature, the identity of Bianchi, Levi-Civita connection. • Geodesics: Definition, local existence, exponential map, minimal geodesics, theorem of Hoph-Rinow, completeness and isometries. • Metric properties of submanifolds of the Riemannian manifolds: normal field, Gauss equation, equations of Weingarton and of Gauss-Codazzi, Egreguim theorem, submanifolds totally geodesic. • Calculus of variations: variations of C -curves, space of paths, critical paths, energy of a path, formula of the first and second variation, Jacobi field, index theorem.