Sumários

Characteristic classes. Fiber bundles.

12 Dezembro 2017, 13:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

Pontryagin classes and Chern classes. Complex vector bundles. Complexification of a real vector bundle. Relation between Pontrjagin end Chern classes.
The notion of fiber bundle with group structure G. Principal G-bundles. Examples.


Characteristic classes.

11 Dezembro 2017, 13:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

De Rham cohomology of M with coefficients in the total space E of a vector bundle over M. Correspondence between homogeneous polynomials of degree  k in a vector space V and k-multilinear symmetric maps. Chern-Weil homomorphism.


Curvature and holonomy

5 Dezembro 2017, 13:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

Curvature tensor and curvature 2-form. Structure equations. Bianchi's identity. Holonomy of a connection. Holonomy representation of a flat connection.
Differential forms in M with values in the total space E of a vector bundle over M.


Connections

4 Dezembro 2017, 13:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

The covariant derivative of a section and parallel transport along a differentiable path.
Applications to the tangent bundle: geodesic, exponential map, torsion. Levi-Civita connection in Riemannian manifolds.


Poincaré-Hopf Theorem. Pull-back bundles.

28 Novembro 2017, 13:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

The relationship between the Euler class of the tangent bundle and the Euler characteristic, and as consequence, the Poincaré-Hopf Theorem.
The pull-back of vector bundles. The notion of connection in a vector bundle. Definition of Christoffel symbols  and 1-forms of the connection.