nLab affine connection

Definition

An affine connection $\nabla$ on a smooth manifold $M$ is a connection on the frame bundle $F M$ of $M$, i.e., the principal bundle of frames in the tangent bundle $T M$.

The components of the local Lie-algebra valued 1-form of an affine connection are called Christoffel symbols.

Related concepts

• connection on a bundle

  • parallel transport, holonomy

• principal connection

  • affine connection, Levi-Civita connection, Cartan connection, symplectic connection

• connection on a 2-bundle

• connection on an infinity-bundle

  • higher parallel transport

References

A coordinate-free treatment first appeared in

• Harley Flanders?, Development of an extended exterior differential calculus. Transactions of the American Mathematical Society 75:2 (1953), 311–311. doi.

• wikipedia affine connection

• eom affine connection

• А.П. Норден, Пространства аффинной связности, 1976, djvu