Connection (mathematics)

The parallel transport of a vector along a closed curve over the sphere, which, like the concept of covariant derivative, is based on the notion of mathematical connection. The angle α α {displaystyle alpha } After going through once the curve is proportional to the area within the curve.

In differential geometry, the connection is a mathematical object defined on a differentiable manifold that allows establishing a relationship or "connecting" the local geometry around a point with the local geometry around another point. The simplest case of connection is an affine connection that allows one to specify a covariant derivative on a differentiable manifold.

Introduction

The theory of connections leads to curvature invariants (see also curvature tensor), and torsion. This applies to tangent bundles; there are more general connections, in differential geometry: a connection can refer to a connection in any vector bundle or to a connection in a main bundle.

In one particular approach, a connection is a 1-form to values in a Lie algebra that is a multiple of the difference between the covariant derivative and the ordinary partial derivative. That is, the partial derivative is not an intrinsic notion in a differentiable manifold: a connection corrects the concept and allows discussion in geometric terms. The connections give rise to a parallel transport.

Connection types

There are a large number of possible approaches related to the concept of connection, among which are the following:

• A very direct module style to the covariant differentiation, indicating the conditions that allow the vector fields to act on sections of vector fibers.
• The traditional notation of specific indexes the connection by the components, sees covariant derivative (three indexes, but this It's not. a tensor).
• In Riemann geometry there is a way to derive a connection of the metric tensor (Levi-Civita connection).
• Using main fibers and differential shapes at values in a Lie algebra (see Cartan connection).
• the most abstract approach can be suggested by Alexander Grothendieck, where it is considered a connection as a descent of 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 common form of this is Schwarz derivative in complex analysis. See also: Gauss-Manin connection