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 topological bundles and fibrations

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, in which case $\pi$ is often called fibration.

Formal definition

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 $r,t,s\in[0,1]$ in any order, $\tau_t^t\equiv\operatorname{id}$,
• 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.

The homeomorphism $\tau_\gamma=\tau_0^1:F_a\to F_b$, $a=\gamma(0)$, $b=\gamma(1)$, is called the parallel transport along the path $\gamma$. By the natural extension, it is defined for closed paths $\gamma$ beginning and ending at any point $a$ as a self-map of the fiber $F_a$.

Differentiability of the connection

In the case where $\pi$ is a smooth bundle (fibration), the natural condition is to require that connections are differentiable maps, differentiably depending on the "transport time".

Covariant derivation

Let $\pi$ be a fibration and $s:B\to E$ a smooth section, a differentiable map selecting a point $x=s(a)$ in each fiber $F_a$. A connection on the bundle $E$ allows to differentiate $s$ along a smooth (or piecewise smooth) path $\gamma$, with the derivative (at the initial point $a$) is a vector tangent to the fiber $F_a$ at the point $s(a)$.

The construction goes as follows: for every $t\in (\R^1,0)$ the parallel transport map $\tau_t^0=(\tau_0^t)^{-1}$ maps the point $s(\gamma(t))\in F_{\gamma(t)}$ back into the fiber $F_a=F_{\gamma(0)}$, defining thus a continuous curve $$ (\R^1,0)\owns t\mapsto v(t)=\tau_t^0(s(\gamma(t))\in F_a. $$ If the parallel transport maps, the section and the transport curve $\gamma$ are all smooth (the condition that should be verified in the local trivializing charts), then the curve $t\mapsto v(t)$ is a smooth parametrized curve in the fiber $F_a$ which has a well-defined velocity vector $v\in T_x F\subseteq T_x E$, $x=s(a)$.

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