Linear connection

In mathematics, more specifically in differential geometry, the term linear connection can refer to either of the following overlapping concepts:

• a connection on a vector bundle, often viewed as a differential operator (a Koszul connection or covariant derivative);
• a principal connection on the frame bundle of a manifold or the induced connection on any associated bundle — such a connection is equivalently given by a Cartan connection for the affine group of affine space, and is often called an affine connection.

The two meanings overlap, for example, in the notion of a linear connection on the tangent bundle of a manifold.