UCL - Differential geometry II

[MAT2420]

[30h + 15h] 5 credits

This course is taught in the 1st semester.

Teacher(s): Pierre Van Moerbeke

Language: French

Level: Second

Aims
Main themes
Other information (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)

Aims

Ability of defining, describing and using on specific examples that are homogeneous spaces of Lie groups basic notions of differential geometry attached to the ones of Riemannian structure and affine connection.

Main themes

Tensor bundles on a manifold, sections, skewsymmetric forms, exterior differential, Riemannian metric, arclenght, Riemannian volume element, geodesic as local solutions of a variational principle, Christoffel symbols and their transformation law, affine connection, Levi-Civita theorem, torsion, curvature, Lie derivative, Cartan formula, geodesic as autoparallel curves, covariant derivatives of tensor fields, normal coordinates, Lie algebra associated to a Lie group, Lie's theorems (introductive), exponantial mapping, action of a Lie group on a smooth manifold, invariant tensor fields, isometries, homogeneous spaces, reductivity and invariant connections, symmetric spaces.

Other information (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)

Precursorycourses Géométrie et topologie différentielles I

Evaluation Written examination.