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:

$\begin{align}&\nabla_X(\sigma_1 + \sigma_2) = \nabla_X\sigma_1 + \nabla_X\sigma_2\\ &\nabla_{X_1 + X_2}\sigma = \nabla_{X_1}\sigma + \nabla_{X_2}\sigma\\ &\nabla_{X}(f\sigma) = f\nabla_X\sigma + X(f)\sigma\\ &\nabla_{fX}\sigma = f\nabla_X\sigma.\end{align}$

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.

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