Connection (vector Bundle) - Formal Definition

Formal Definition

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ(E). A connection on E is an ℝ-linear map

such that the Leibniz rule

holds for all smooth functions f on M and all smooth sections σ of E.

If X is a tangent vector field on M (i.e. a section of the tangent bundle TM) one can define a covariant derivative along X

by contracting X with the resulting covariant index in the connection ∇ (i.e. ∇_Xσ = (∇σ)(X)). The covariant derivative satisfies the following properties:

Conversely, any operator satisfying the above properties defines a connection on E and a connection in this sense is also known as a covariant derivative on E.