Differential geometry MOC

Affine connexion

An affine connexion1 is additional2 structure on a -manifold which connects nearby tangent spaces so as to enable Parallel transport as in an affine space. #m/def/geo/diff With an affine connexion one can define

An affine connexion is not unique, the disagreement between two connexions is described by the Connexion disagreement tensor.

As a differential operator

An affine connexion is an -linear map

from vector fields to -tensor fields which satisfies a Leibniz rule

where is the exterior derivative. We write

for . This can then be extended to all tensor fields as the Covariant derivative.

Examples

#state/develop | #lang/en | #SemBr

Footnotes

  1. The only reason I spell it this way is because I think it's fun. ↩

  2. In some cases other structure on the manifold provides a canonical choice of connexion, e.g. a semi-Riemannian metric gives the Levi-Civita connexion. ↩