Editing Connection (vector bundle) (section)

==Examples==

* A classical [[covariant derivative]] or [[affine connection]] defines a connection on the [[tangent bundle]] of ''M'', or more generally on any [[tensor bundle]] formed by taking tensor products of the tangent bundle with itself and its dual.
* A connection on <math>\pi: \R^2\times \R \to \R</math> can be described explicitly as the operator
::<math>\nabla = d + \begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix}dx</math>
:where <math>d</math> is the exterior derivative evaluated on vector-valued smooth functions and <math>f_{ij}(x)</math> are smooth. A section <math>a \in \Gamma(\pi)</math> may be identified with a map 
::<math>\begin{cases} \R \to \R^2 \\ x \mapsto (a_1(x),a_2(x)) \end{cases}</math>
:and then
::<math>\nabla(a)= \nabla \begin{bmatrix} a_1(x) \\ a_2(x) \end{bmatrix} = \begin{bmatrix} \frac{da_1(x)}{dx} + f_{11}(x)a_1(x) + f_{12}(x)a_2(x) \\ \frac{da_2(x)}{dx} + f_{21}(x)a_1(x) + f_{22}(x)a_2(x)\end{bmatrix}dx</math>
* If the bundle is endowed with a [[bundle metric]], an inner product on its vector space fibers, a [[metric connection]] is defined as a connection that is compatible with the bundle metric.
* A [[Yang-Mills connection]] is a special [[metric connection]] which satisfies the [[Yang-Mills equation]]s of motion.
* A [[Riemannian connection]] is a [[metric connection]] on the tangent bundle of a [[Riemannian manifold]].
* A [[Levi-Civita connection]] is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also [[torsion tensor|torsion-free]]. It is unique, in the sense that given any Riemannian connection, one can always find one and only one equivalent connection that is torsion-free.  "Equivalent" means it is compatible with the same metric, although the curvature tensors may be different; see [[teleparallelism]]. The difference between a Riemannian connection and the corresponding Levi-Civita connection is given by the [[contorsion tensor]].
* The [[exterior derivative]] is a flat connection on <math>E=M \times \R</math> (the trivial line bundle over ''M'').
* More generally, there is a canonical flat connection on any [[flat vector bundle]] (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.