In one particular approach, a connection is a Lie algebra valued 1-form which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative. That is, *partial derivatives* are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms. Connections give rise to parallel transport.

There are quite a number of possible approaches to the connection concept. They include the following:

- A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act on vector bundle sections.
- Traditional index notation specifies the connection by components (three indices, but this is
a tensor).*not* - In Riemannian geometry there is a way of deriving a connection from the metric tensor (Levi-Civita connection).
- Using principal bundles and Lie algebra-valued differential forms (see Cartan connection).
- The most abstract approach may that suggested by Alexander Grothendieck, where a connection is seen as descent data from infinitesimal neighbourhoods of the diagonal.