Affine connexion

An affine connexion¹ is additional² structure on a $𝐶 𝛼$ -manifold $(𝑀, 𝒜)$ 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 $\nabla$ is an $ℝ$ -linear map

𝔛 (𝑀) \to T 11 (𝑀) 𝑋 𝑎 \mapsto \nabla 𝑏 𝑋 𝑎

from vector fields to $(1, 1)$ -tensor fields which satisfies a Leibniz rule

\nabla 𝑏 (𝑓 𝑋 𝑎) = (d 𝑓) 𝑏 𝑋 𝑎 + 𝑓 \nabla 𝑏 𝑋 𝑎

where $(d 𝑓) 𝑏$ is the exterior derivative. We write

\nabla 𝑌 𝑋 𝑎 : = 𝑌 𝑏 \nabla 𝑏 𝑋 𝑎 \in 𝔛 (𝑀)

for $𝑌 𝑎 \in 𝔛 (𝑀)$ . This can then be extended to all tensor fields as the Covariant derivative.

Examples

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

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

Affine connexion

As a differential operator

Examples

Footnotes