Connection (mathematics) - Possible Approaches

Possible Approaches

• A rather direct approach is to specify how a covariant derivative acts on elements of the module of vector fields as a differential operator. More generally, a similar approach applies for connections in any vector bundle.
• Traditional index notation specifies the connection by components; see Christoffel symbols. (Note: this has three indices, but is not a tensor).
• In pseudo-Riemannian and Riemannian geometry the Levi-Civita connection is a special connection associated to the metric tensor.
• These are examples of affine connections. There is also a concept of projective connection, of which the Schwarzian derivative in complex analysis is an instance. More generally, both affine and projective connections are types of Cartan connections.
• Using principal bundles, a connection can be realized as a Lie algebra-valued differential form. See connection (principal bundle).
• An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is the Ehresmann connection.
• The most abstract approach may be that suggested by Alexander Grothendieck, where a Grothendieck connection is seen as descent data from infinitesimal neighbourhoods of the diagonal; see (Osserman 2004).