A connection on a vector bundle 
is a way to "differentiate" bundle
sections, in a way that is analogous to the exterior
derivative 
of a function 
.
In particular, a connection 
is a function from smooth sections 
to smooth sections of 
with one-forms 
that satisfies the following conditions.
1. 
(Leibniz rule), and
2. 
.
Alternatively, a connection can be considered as a linear map from bundle sections of 
, i.e., a section of 
with a vector field 
, to sections of 
, in analogy to the directional
derivative. The directional derivative
of a function 
,
in the direction of a vector field 
, is given by 
. The connection, along with a vector field 
, may be applied to a section 
of 
to get the section 
. From this perspective, connections must also satisfy
 |
(1)
|
for any 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.
The matrix of one-forms
![alpha=[dx 2xdy 0; 0 dx-3dy 0; xydx 0 y^2dx+dy]](/images/equations/VectorBundleConnection/NumberedEquation2.svg) |
(2)
|
determines a connection 
on the rank-3 bundle over 
. It acts on a section 
by the following.
 |  |  |
(3)
|
 |  | ![s_x+[1 0 0; 0 1 0; xy 0 y^2]s](/images/equations/VectorBundleConnection/Inline38.svg) |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  | ![s_y+[0 2x 0; 0 -3 0; 0 0 1]s](/images/equations/VectorBundleConnection/Inline47.svg) |
(7)
|
 |  |  |
(8)
|
In any trivialization, a connection can be described just as in the case of a trivial bundle. However,
if the 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 bundle curvature of the bundle is given by the formula 
.
In coordinates, 
is matrix of two-forms.
For instance, in the example above,
![Omega=[0 2xdx ^ dy 0; 0 -3dx ^ dy 0; 0 2x^2ydx ^ dy y^2dx ^ dy]](/images/equations/VectorBundleConnection/NumberedEquation3.svg) |
(9)
|
is the curvature.
Another way of describing a connection is as a splitting of the tangent bundle 
of 
as 
.
The vertical part of 
corresponds to tangent vectors along the fibers, and is the
kernel of 
.
The horizontal part is not well-defined a priori.
A connection defines a subspace of 
which is isomorphic to 
. It defines 
flat sections 
such that 
, which are a vector basis
for the fiber bundles of 
, at least nearby 
. These flat sections determine the horizontal part of 
near 
.
Also, a connection on a vector bundle can be defined by a principal
bundle connection on the associated
principal bundle.
In some settings there is a canonical connection. For example, a Riemannian manifold has the Levi-Civita connection, given by the Christoffel symbols of the first and second kinds, which is the unique torsion-free connection compatible with the metric. A holomorphic vector bundle with a Hermitian metric has a unique connection which is compatible with both metric and the complex structure.
