Sumários
Transformation groups
23 outubro 2017, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Transformation groups: concrete realization of Lie groups. Definition of proper action. Smooth structure on the quotient space by a free and proper action.
The orbits of a smooth action are (immersed) submanifolds.
Integration of Lie algebras homomorphisms. Exponential map
17 outubro 2017, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Integration of Lie algebras homomorphisms to Lie groups homomorphisms. Definition of exponential map and its properties.
Integration of Lie algebras
16 outubro 2017, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Lie subalgebras and Lie subgroups. Examples. Integrations of Lie algebras to Lie groups.
Frobenius Theorem. Lie groups and Lie algebras
10 outubro 2017, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Involutive and integrable distributions. Frobenius Theorem: a distribution is integrable iff it is involutive, that is, foliations are the global objects associated with involutive distributions.
Definition of Lie group and Lie algebra. Examples. Left invariant vector fields. Examples. Homomorphism of Lie algebras and homomorphism of Lie groups.
Lie bracket. Distributions
9 outubro 2017, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Definition of Lie bracket and its properties. Relation of the Lie bracket with the flows of the vector fields. Definition of Lie derivative and its relation with the Lie bracket. Definition of distribution. Examples. Integral manifolds.