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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the derivative is one of the two central concepts of calculus. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...

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. In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... Curvature is the amount by which a geometric object deviates from being flat. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...

General concept

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 sum 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. In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In 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. ...

Given a the induced bundle has an induced connection. If B' = I is a segment then the 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 connection on a Riemannian manifold is called parallel transport). In 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). ... In 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. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

There are many ways to describe a 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 derivative and the ordinary partial derivative 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. (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In 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. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

Possible approaches

There are quite a number of possible approaches to the connection concept. They include the following:

• A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections.
• Traditional index notation specifies the connection by components; see Covariant derivative (three indices, but this is not a tensor).
• In Riemannian geometry there is a way of deriving a connection from the metric tensor (Levi-Civita connection).
• Using principal bundles and Lie algebra-valued differential forms (see connection form and Cartan connection).
• The most abstract approach may be that suggested by Alexander Grothendieck, where a connection is seen as descent data from infinitesimal neighbourhoods of the diagonal.

The connections referred to above are linear or affine connections. There is also a concept of projective connection; the most commonly-met form of this is the Schwarzian derivative in complex analysis. In abstract algebra, a module is a generalization of a vector space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In 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). ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...

See also: Gauss-Manin connection