Sumários
Vector bundle - VII
23 Abril 2024, 10:00 • Giordano Cotti
In this lecture, we constructed two distinct (i.e., non-isomorphic) complex line bundle structures on the real tangent bundle of the sphere . We explicitly demonstrated that these complex line bundles are dual to each other. Finally, we formulated the "hairy ball theorem" and provided a complete proof, following the paper by J. Milnor [Milnor, American Mathematical Monthly, July 1978, 521-524].
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.