Connection (composite bundle)

A connection in mathematics, particularly differential geometry and related fields, refers to a structure defined on a fiber bundle that allows for the differentiation of sections along tangent vectors. In the context of a composite bundle, the notion of a connection becomes more nuanced, reflecting the layered or hierarchical structure of the bundle.

A composite bundle, sometimes also referred to as a fibered manifold over another fibered manifold, can be described as a tower of projections: E → F → M, where E, F, and M are manifolds, and the arrows denote fiber bundle projections. This implies that for each point in M, there's a fiber F over it, and for each point in F, there's a fiber E over it. In other words, E is a fiber bundle over F, which is, in turn, a fiber bundle over M.

A connection on a composite bundle addresses how to "connect" or "transport" data (e.g., tangent vectors, forms) along curves within this multi-layered structure. The connection defines how to compare fibers at different points in the base manifold (M or F, depending on the specific layer considered). It essentially provides a way to parallel transport sections of the bundle.

Crucially, when dealing with composite bundles, there are different types of connections that can be defined, depending on which projection is being considered. One might consider a connection on E → F, a connection on F → M, or a more sophisticated connection that incorporates information from both projections.

A connection on E → F allows one to differentiate sections of E along tangent vectors in F. A connection on F → M allows one to differentiate sections of F along tangent vectors in M. These connections can be independent, or they can be related by compatibility conditions depending on the context and the specific problem being addressed.

In general, a connection can be defined through various formalisms, such as:

• Horizontal Subspaces: A splitting of the tangent space of the total space (E or F) into a horizontal subspace (tangent to the base manifold) and a vertical subspace (tangent to the fiber).

• Connection Forms: Differential forms on the total space with values in the Lie algebra of the structure group of the bundle.

• Covariant Derivatives: Operators that extend the notion of differentiation to sections of the bundle.

The choice of formalism depends on the specific application and the properties that are desired. The concept of a connection in a composite bundle is crucial in areas like gauge theory, general relativity (where spacetime itself can be viewed as a composite bundle), and the study of higher-order differential equations. The structure allows for the consistent definition of derivatives and parallel transport within the complex geometry described by the composite bundle.