Editing Differential geometry (section)

== Bundles and connections ==

The apparatus of [[vector bundle]]s, [[principal bundle]]s, and [[connection (mathematics)|connection]]s on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the [[tangent bundle]]. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of [[parallel transport]]. An important example is provided by [[affine connection]]s. For a surface in '''R'''<sup>3</sup>, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In [[Riemannian geometry]], the [[Levi-Civita connection]] serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be [[spacetime]] and the bundles and connections are related to various physical fields.