Dual connection
From Diffgeom
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: