Associated fibre bundles - Lec 20 - Frederic Schuller

The video begins with an introduction to associated bundles, which are mathematical constructs that provide a deeper understanding of principal fiber bundles. These are discussed in the context of physics, emphasizing their significance in the field. The concept of a principal bundle is first reviewed, followed by the precise definition of an associated bundle, which involves a G-principal bundle, a smooth manifold F, and a left G-action on F. The construction of the associated bundle involves an equivalence relation and the creation of a new bundle space P_F with a projection map π_F.

The script explains the process of defining an equivalence relation on the Cartesian product of the principal bundle's total space and the manifold F. This relation identifies elements based on the action of the group G. The quotient space formed by this relation is denoted as P_F and is considered to be the total space of a new bundle with the base space M. The fibers of this bundle are tied to the principal bundle through the action of G.

The paragraph discusses the construction of the associated projection map π_F, which is well-defined due to the properties of the equivalence relation. The map takes an element of the associated bundle, viewed as an equivalence class, and projects it to a point in M using the principal bundle's projection map π. The script also addresses the need to check the map's well-definedness by considering different representatives of the equivalence class.

The video provides examples of associated bundles, starting with the frame bundle, which is a principal bundle where each point of the manifold has a basis of the tangent space. The script then connects this to the concept of the tangent bundle as an associated bundle with the fiber being R^D. The left action of GL(D, R) on the fiber is defined, and the resulting associated bundle is shown to be isomorphic to the tangent bundle.

The script explains how the components of elements in the tangent space transform when the frame is changed by a general linear transformation. This transformation is imposed when constructing an associated bundle, and the script emphasizes that this choice of transformation is what physicists often refer to when discussing how components change under a frame change.

The video claims the existence of a bundle isomorphism between the newly constructed associated bundle and the known tangent bundle. This isomorphism is defined by a map that takes an element of the associated bundle and sends it to an element of the tangent bundle. The components of the fiber element transform in the opposite direction under the group action, which is a key concept in physics.

The script introduces the concept of tensor densities, which are defined by modifying the left action of the group on the fiber to include an additional factor, the determinant of the group element's inverse raised to a power Omega. This modification results in a new type of bundle called the tensor density bundle. The script also differentiates between tensor bundles and tensor density bundles.

The video defines scalar densities as a special case of tensor densities where the fiber is kept invariant and only scaled by the determinant of the group element's inverse. The script explains that scalar densities are important in situations where integration over a manifold is required, as they allow for well-defined integrals under changes of coordinates.

The script connects the concept of densities to general relativity, where the metric tensor transforms as a density under general linear group transformations. This transformation property is crucial for understanding the Einstein action and the integration of Lagrangian densities in the theory.

The video defines associated bundle maps, which are bundle maps constructed from a principal bundle map between the underlying principal bundles. The script explains the conditions for such a map to exist and how it accounts for the specific structure of associated bundles.

The script discusses the concept of triviality for associated bundles, explaining that an associated bundle is considered trivial if the underlying principal bundle is trivial. The script also highlights the difference between a trivial fiber bundle and a trivial associated bundle, noting that the latter carries more structure.

The video introduces the idea of restricting and extending bundles with respect to different underlying principal bundles. The script defines when a principal bundle can be restricted to a subgroup or extended from a closed subgroup and discusses the conditions under which these operations are possible.

The script presents a theorem that determines whether a principal bundle can be restricted to a subgroup based on the existence of a cross section for the quotient bundle. The video provides examples related to the frame bundles and their ability to be equipped with different metrics, such as Riemannian or Lorentzian, and discusses the implications for the structure of the manifold.

The video concludes with a discussion on the limitations of Lorentzian manifolds and the conditions under which they can support spin structures. The script highlights that not all manifolds can carry a spin structure, even if they admit a Lorentzian metric, which is an important consideration in the study of fermions in the context of general relativity.