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
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
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
determines a connection on the rank-3 bundle over . It acts on a section by the following.
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,
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.