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.

Read more about this topic: Connection (vector Bundle)

Famous quotes containing the words formal and/or definition:

“There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
—Sara Lawrence Lightfoot (20th century)

“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)