## Definition

Suppose is a vector bundle over a differential manifold and is a connection on . The **dual connection** to , denoted , is a connection on the dual vector bundle , defined as follows.

For any and , we have:

where

## Motivation

The definition of a dual connection is chosen in such a way that the bilinear form for evaluation:

satisfies the Leibniz rule. In other wors, we need to ensure that for and , we have: