Parallel transport

A topological or differential geometric construction generalizing the idea of parallel translation in affine spaces to general bundles. In contrast with the affine case, the result of parallel transport along a closed path may in general be nontrivial, leading thus to the notion of curvature.

Parallel transport (translation) in affine spaces

If $A$ is an affine space associated with the vector space $V=\Bbbk^n$ (over the field $\Bbbk$, usually $\Bbbk=\R$), then $V$ acts on $A$ by parallel translations $\{t_w:w\in V\}$: $$ \forall x=(a_1,\dots,a_n)\in A^n,\ \forall w=(w_1,\dots,w_n)\in V\qquad t_w x=(a_1+w_1,\dots,a_n+ w_n). $$ This action induces the (almost trivial) action of parallel transport on tangent vectors. If $TA\simeq V\times A\simeq\Bbbk^{2n}=\{(v,a)\}$ is the tangent bundle, the collection of vectors $v$ attached to different points $a\in A$, then the parallel transport acts on $TA$ by its differential, $$ \forall v\in T_aA,\ \forall w\in V,\qquad \rd t_w(a)\cdot v=v\in T_{t_w(a)}=T_{a+w} A. $$ Consequently, if $w_1,\dots,w_k\in V$ are vectors such that $w=w_1+\cdots+w_k=0$, then the action $t_{w_k}\circ\cdots\circ t_{w_1}:T_a A\to T_a A$ is the identity for any point $a$.

These trivial observations indicate some of the properties that will fail for general parallel transport.

Parallel transport in bundles: informal definition

Let $\pi:E\to B$ be a topological bundle with a generic fiber $F$, with all three topological spaces eventually having some additional structures defined on them. Usually we will assume that $E,B,F$ are smooth (finite-dimensional) manifolds with $\pi$ a differentiable map of full rank.

A connection in the topological bundle is a correspondence which allows to associate with any simple path $\gamma:[0,1]\to B$ in the base a family of homeomorphisms $\tau|_t^s:\pi^{-1}(\gamma(t))\to\pi^{-1}(\gamma(s))$ between the respective fibers $F_t=\pi^{-1}(\gamma(t))$ and $F_s=\pi^{-1}(\gamma(s))$ such that:

• $\tau|_t^s\circ \tau|_r^t=\tau|_{r}^s$ for all values $r,t,s\in[0,1]$,
• the homeomorphisms $\tau|_{t}^s$ continuously depend on $t,s\in[0,1]$,
• the homemorphisms $\tau|_t^s$ preserve the additional structure on the fibers, if any.

E.g., if all fibers are linear or Euclidean spaces, then $\tau|_t^s$ must be linear, resp., linear orthogonal operators.