information:filtering
Connections

An (affine) connection is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

Let X (M) {\displaystyle {\mathfrak {X}}(M)} denote the space of vector fields on M {\displaystyle M}. An (affine) connection

on M {\displaystyle M} is a bilinear map (X, Y) ↦ ∇ X Y {\displaystyle (X,Y)\mapsto \nabla _{X}Y} such that

The expression ∇ X Y {\displaystyle \nabla _{X}Y} is called the covariant derivative of Y {\displaystyle Y} with respect to X {\displaystyle X}.