A connection on a vector bundle 
is an operation 
that assigns to a vector field 
and a smooth section 
a section 
, is linear over smooth functions
in 
, is linear over the scalar field
in 
, and satisfies 
. It defines parallel
transport and covariant differentiation.
Its failure to commute in two directions is measured by the bundle
curvature.
Connection
See also
Bundle Curvature, Covariant Derivative, Levi-Civita Connection, Vector Bundle ConnectionExplore with Wolfram|Alpha
References
Kobayashi, S. and Nomizu, K. Foundations of Differential Geometry, Vol. 1. New York: Wiley, 1963.Cite this as:
Weisstein, Eric W. "Connection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Connection.html