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. diff With an affine connexion one can define

• Parallel transport
• Covariant derivative
• Torsion tensor
• Riemannian curvature

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

• Partial derivative as a local affine connexion
• Levi-Civita connexion

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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. ↩