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.