Parallel Transport - Parallel Transport On A Vector Bundle

Parallel Transport On A Vector Bundle

Let M be a smooth manifold. Let E→M be a vector bundle with covariant derivative ∇ and γ: I→M a smooth curve parameterized by an open interval I. A section of along γ is called parallel if

Suppose we are given an element e₀ ∈ E_P at P = γ(0) ∈ M, rather than a section. The parallel transport of e₀ along γ is the extension of e₀ to a parallel section X on γ. More precisely, X is the unique section of E along γ such that

Note that in any given coordinate patch, (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–Lindelöf theorem guarantees the existence and uniqueness of the solution.

Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve:

from the vector space lying over γ(s) to that over γ(t). This isomorphism is known as the parallel transport map associated to the curve. The isomorphisms between fibers obtained in this way will in general depend on the choice of the curve: if they do not then parallel transport along every curve can be used to define parallel sections of E over all of M. This is only possible if the curvature of ∇ is zero.

In particular, parallel transport around a closed curve starting at a point x defines an automorphism of the tangent space at x which is not necessarily trivial. The parallel transport automorphisms defined by all closed curves based at x form a transformation group called the holonomy group of ∇ at x. There is a close relation between this group and the value of the curvature of ∇ at x; this is the content of the Ambrose-Singer holonomy theorem.