A vector bundle connection on a vector bundle 
is a way to "differentiate" bundle sections,
analogous to the exterior derivative 
of a function 
. In particular, a connection is a linear
transformation
 |
(1)
|
that satisfies the following conditions.
1. 
(product rule), and
2. 
.
Equivalently, a connection assigns to each vector field 
a covariant derivative 
from bundle sections
to bundle sections. This is analogous to the directional derivative 
of a function 
in the direction 
. From this perspective, a connection must satisfy
 |
(2)
|
for every smooth function 
. This property follows from the first definition.
For example, the trivial bundle 
admits a flat connection since any bundle
section 
corresponds to a function 
. Then setting 
gives the connection. Any connection on the trivial
bundle is of the form 
, where 
is any one-form with values
in 
,
i.e., 
is a matrix of one-forms.
![alpha=[dx 2xdy 0; 0 dx-3dy 0; xydx 0 y^2dx+dy]](/images/equations/VectorBundleConnection/NumberedEquation3.svg) |
(3)
|
determines a connection 
on the rank-3 vector
bundle over 
. It acts on a bundle section 
by the following.
 |  |  |
(4)
|
 |  | ![s_x+[1 0 0; 0 1 0; xy 0 y^2]s](/images/equations/VectorBundleConnection/Inline28.svg) |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  | ![s_y+[0 2x 0; 0 -3 0; 0 0 1]s](/images/equations/VectorBundleConnection/Inline37.svg) |
(8)
|
 |  |  |
(9)
|
In any trivialization, a connection can be described just as in the case of a trivial bundle. However,
if the vector bundle 
is not trivial, then the
exterior derivative 
is not well-defined (globally)
for a bundle section 
. Still, the difference between any two connections must be
one-forms with values in endomorphisms
of 
,
i.e., matrices of one-forms.
So the space of connections forms an affine
space.
The local connection matrix depends on the choice of frame. Write a frame as a row 
, so a bundle section
is 
for a column 
of smooth functions, and 
. If the new frame is 
, where 
is a smooth invertible matrix-valued
function, then 
and
 |
(10)
|
The bundle curvature instead transforms by
 |
(11)
|
Thus the connection matrix has an extra derivative term, whereas the bundle curvature transforms as a 2-form with values in endomorphisms of the vector bundle (Merry 2007, pp. 51-53).
The connection extends to an exterior covariant derivative on 
-valued differential forms,
also denoted 
.
On bundle sections, its square is multiplication
by the bundle curvature 
, so that 
. In coordinates,
is a matrix of 2-forms. For
instance, in the example above,
![Omega=[0 2 0; 0 0 0; -x(1+y) 2x^2y -2y]dx ^ dy](/images/equations/VectorBundleConnection/NumberedEquation6.svg) |
(12)
|
is the curvature.
Another way of describing a connection is as a horizontal subspace 
complementary to the vertical subspace 
at every 
, so that
 |
(13)
|
The restriction of the tangent map 
to 
is an isomorphism onto 
.
For a vector bundle connection, these horizontal
subspaces vary smoothly and compatibly with the linear
structure of the fibers.
In some settings there is a canonical connection. For example, the tangent bundle of a Riemannian manifold has the Levi-Civita connection, which is the unique torsion-free connection compatible with the Riemannian metric. A holomorphic vector bundle with a Hermitian metric has a unique connection compatible with both the Hermitian metric and the holomorphic structure.
