Sumários

Vector bundles - VI

17 Abril 2024, 15:30 • Giordano Cotti

In this lecture:


  • we proved that if a complex vector bundle admits a real form (i.e. it is the complexification of a real vector bundle), then it is canonically isomorphic to its dual E*;
  • we mentioned the existence of counterexamples of the previous statement (complex vector bundle E with no real form and isomorphic to E*);
  • we discussed the surjectivity issue of the "complexification" map, from the set of real vector bundles to the set of complex vector bundles;
  • in particular, we claimed and proved that any real line bundle on the sphere S^2 is trivial; we also claimed the existence of non-trivial complex line bundle on S^2.

Vector bundles - V

16 Abril 2024, 10:00 • Giordano Cotti

In this lecture:


  • we showed the existence of a smooth metric on any given real vector bundle E on M, by using a partition of unity construction;
  • we introduced a group structure on the set of isomorphism classes of complex line bundles on a smooth manifold M (complex smooth Picard group);
  • we introduced the notion of conjugate bundle of a complex vector bundle;
  • we showed that, given a complex vector bundle E, its conjugate is canonically isomorphic to its dual bundle E*;
  • we introduced the notion of hermitian metric on a complex vector bundle, and proved its existence via a partition of unity construction;
  • we introduced the notion of complexification of a given real vector bundle.

Vector bundles - IV

10 Abril 2024, 15:30 • Giordano Cotti

In this lecture:

  • we discussed the classification problem of real line bundles on the real n-dimensional projective space;
  • we introduced the notions of global/local sections of a vector bundle;
  • we introduced the notion of a module over a ring, and realized that the spaces of local sections of a vector bundle on M are naturally equipped with a module structure over the ring of smooth functions on M;
  • we introduced the notions of tensor field on a manifold M, and mentioned distinguished examples (e.g. vector fields, k-forms, polyvector fields);
  • we introduced the notion of a local frame of a vector bundle E on M;
  • we showed that the datum of a local frame of E is equivalent to a local trivialization of E;
  • we deduced a criterion for triviality of a vector bundle: E is trivial if and only if it admits a global frame; in particular, a line bundle is trivial iff it admits a nowhere vanishing section;
  • we showed that the Möbius strip, seen as a vector bundle over S^1, is not trivial;
  • we introduced a group structure on the set of isomorphism classes of real line bundles on M, the operation being the tensor product (real smooth Picard group of M);
  • we explained why each element of the real smooth Picard group of M has order two;
  • More generally, we explained that a real vector bundle E on M is isomorphic to its dual E*.

Vector bundles - III

9 Abril 2024, 10:00 • Giordano Cotti

In this lecture we introduced two further operations on vector bundles, namely the symmetric and exterior powers. 

These bundles have been introduced by prescribing the corresponding systems of transition functions.

A problem has been assigned: classification of real line bundles on the n-dimensional real projective space up to isomoprhisms.

Vector bundles - II

27 Março 2024, 15:30 • Giordano Cotti

In this lecture:

  • we introduced the notion of morphisms of vector bundles;
  • isomorphism classification problem for vector bundles: we prove that two vector bundles on a same smooth manifolds are isomorphic if and only if they admit systems of transition functions which are "cobordant";
  • we recalled basic examples of vector bundles (trivial, tangent, cotangent bundles);
  • we introduced several operations of vector bundles: dualization, Whitney direct sum, tensor product. These operations allow to construct new bundles from given ones.