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In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. That is an application to tangent bundles; there are more general connections, used in differential geometry and other fields of mathematics to formulate intrinsic differential equations . Connection may refer to a connection on any vector bundle, or also a connection on a principal bundle.
Connections give rise to parallel transport along a curve on a manifold. A connection also leads to invariants of curvature (see also curvature tensor and curvature form), and the so-called torsion.
The general concept can be summarized as follows: given a fiber bundle the tangent space at any point of E has canonical "vertical" subspace, the subspace tangent to the fiber. The connection fixes a choice of "horizontal" subspace at each point of E so that the tangent space of E is a direct sumIn abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vector spaces is the "most general" vector space that contains the of vertical and horizontal subspaces. Usually more requirements are imposed on the choice of "horizontal" subspaces, but they depend on the type of the bundle.
Given a the induced bundle has an induced connection. If is a segment then connection on B gives a trivialization on the induced bundle over I, i.e. a choice of smooth one-parameter family of isomorphisms between the fibers over I. This family is called parallel displacement along the curve and it gives an equivalent description of connection (which in case of Levi-Civita connectionIn Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The Fundamental theorem of Riemannian geometry stat on a Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows one to define of various notions as the length is called parallel transport).
There are many ways to describe connection, in one particular approach, a connection can be locally described as a matrix of 1-forms which is the multiplant of the difference between the covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. There is no real difference between the covariant derivative and the connection concept except for the style in which they are introduced. and the ordinary partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. They are useful in n-dimensional calculus and differential geometry. The partial derivative of in a coordinate chart. That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms.