Connections - V

14 Maio 2024, 10:00 • Giordano Cotti

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In this lecture we introduced the notion of pull-back of vector bundles and connections.

Connections - IV

8 Maio 2024, 15:30 • Giordano Cotti

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In this lecture, we introduced the notion of a flat bundle, which is a bundle that admits at least one flat connection. We proved that a vector bundle is flat if and only if it is possible to choose trivializations with constant transition functions.

We also claimed that on a simply connected manifold $M$, a vector bundle is flat if and only if it is trivial.

Finally, we discussed the twisted de Rham complex with values in a flat vector bundle $E$ equipped with a flat connection.

As an example, we considered the case of the flat bundle given by the orientation line bundle and the corresponding twisted differential forms. In particular, we briefly discussed the top-dimensional case of densities and their integration theory.

Connections - III

7 Maio 2024, 10:00 • Giordano Cotti

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In this lecture, we showed that curvature can be understood as a measure of the extent to which two covariant derivatives along two vector fields do not commute.

Moreover, we discussed gauge transformations for connection 1-forms.

Finally, we demonstrated that curvature can be interpreted as an obstruction to the existence of a local frame of parallel sections.

Connections - II

30 Abril 2024, 10:00 • Giordano Cotti

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In this lecture, we provided a matricial description of a connection and its curvature by introducing a connection 1-form and a curvature 2-form.

We also proved the Bianchi identity, expressing the exterior derivative of the curvature 2-form in terms of the connection 1-form and the curvature 2-form itself.

Connections - I

24 Abril 2024, 15:30 • Giordano Cotti

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In this lecture, we introduced the notion of a connection on a real/complex vector bundle $E$ and the concept of the covariant derivative of a section of $E$ along a vector field.

We demonstrated that connections always exist on a given vector bundle and that the space of connections is an affine space modeled over the infinite-dimensional vector space of 1-forms with values in $\text{End}(E)$.

Given a connection on $E$, we showed how to extend the connection to the entire tensor algebra of $E$.

We also introduced an associated exterior covariant derivative on forms with values in $E$, given a connection on $E$. Finally, we discussed the obstruction for the composition of exterior covariant derivatives to be zero, and we introduced the notion of the curvature of the connection.