Sumários

Principal bundles

18 Dezembro 2018, 11:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

The notion of fibre bundle with group structure G. Morphism of fibre bundles with group structure G.  

Principal G-bundles. The associated G-bundle with fibre F. The frame bundle. 
Connections on a principal bundle. Example of a connection on a frame bundle. 


Characteristic classes (cont.)

17 Dezembro 2018, 11:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

The Chern-Weil homomorphism. Pontryagin classes. 

Complex vector bundles. Chern classes. Complexification of a real vector bundle. 
Relation between Pontrjagin end Chern classes. 


Characteristic classes.

11 Dezembro 2018, 11:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

Differential forms in M with values in the total space E of a vector bundle over M. 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. 


Curvature and holonomy

10 Dezembro 2018, 11:00 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. 

Examples


Connections

4 Dezembro 2018, 11:00 Sílvia Nogueira da Rocha Ravasco dos Anjos

The notion of connection in a vector bundle. Definition of Christoffel symbols  and 1-forms of the connection. The covariant derivative of a section and parallel transport along a smooth path.  

Applications to the tangent bundle: geodesic, exponential map, torsion. Levi-Civita connection in Riemannian manifolds.