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:10.1090/s0002-9947-1953-0057005-8]

See also:

• Wikipedia: Affine connection

• eom: affine connection

Formulation in synthetic differential geometry:

• Wolfgang Bertram; section 10–11 in: Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, Memoirs of the American Mathematical Society 192 (2008) [doi:10.1090/memo/0900, arXiv:math/0502168]

• Anders Kock; section 2.3-4 of: Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010) [pdf, doi:10.1017/CBO9780511691690]

See also:

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