Understanding and Demonstrating the Connection of an Associated Vector Bundle in Differential Geometry

In the field of differential geometry, the concept of a vector bundle is fundamental. A vector bundle E → B is a topological space that is locally a product of a base space B and a vector space. The notion of a connection on a vector bundle is crucial for defining how vector fields on a manifold behave under parallel transportation. This article will delve into the construction of a connection for an associated vector bundle and understand the process through an example.

The Structure of Vector Bundles and Connections

A key component in defining a vector bundle is the concept of local trivializations. This means that for any neighborhood U of the base space B, the restriction of the vector bundle to U can be expressed as a product U × V, where V is a vector space. To make connections between different parts of the bundle, one must ensure that the transitions functions (that describe how different trivializations connect) are smooth and linear.

Constructing a Connection for a Vector Bundle

Theorem 2.21 Every vector bundle E → B admits a connection. This means that there exists a well-defined covariant derivative for sections of E. To construct such a connection, we can use a partition of unity.

Example Construction

Let Wα be an open covering of B by trivializing neighborhoods for E, and Φα the corresponding local trivializations. On each restriction E|Wα, we can consider a trivial product connection dα defined using Φα. However, the expression dαs may only make sense over all of B if the section s ∈ ΓE is zero away from Wα.

Next, consider a partition of unity ρi subordinate to Wα. The expressions ρis ρidαs make sense over all of B since we can extend by zero away from Wi. Now, define E s : X∞i1 dαρis X∞i1 ρidαs for 2.22. The equality here uses the Leibniz rule for dα and the property X∞i1 ρi 1 so X∞i1 dρi 0. This E defined by 2.22 is manifestly linear in s, and the Leibniz rule for E holds because it does for each dα.

Application in Three-Dimensional Descriptive Geometry

Refer to the book Fundamentals of Three-Dimensional Descriptive Geometry by Steve M. Slaby for a more detailed and visual explanation of vector bundles in a three-dimensional context. The chapter on “Principles of Descriptive Geometry Applied to Three-Dimensional Space Vectors” provides easy-to-follow diagrams to help illustrate the concepts discussed here.

Key Takeaways

Local Trivializations: These are the local representations of the vector bundle as U × V, where U is a neighborhood in the base space and V is a vector space. Partition of Unity: This is a collection of functions ρi that sum to one and are used to ensure the constructed connection is well-defined over the entire base space. Leibniz Rule: This is a crucial property used in the construction of the connection, ensuring that the action of the covariant derivative on a product of a section and a smooth function is correctly defined.

In summary, the construction of a connection for a vector bundle involves defining local trivializations and using a partition of unity to extend these trivializations over the entire base space. The Leibniz rule ensures that the resulting connection is well-defined and satisfies the necessary properties.