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 $F_a$.

Smooth connections on fibrations

In the case where $\pi$ is a smooth bundle (fibration), it is natural to require that connections are differentiable maps between the fibers, differentiably depending also on the "transport time". This leads to the infinitesimal (tangent) construction known as the Ehresmann connection.

Covariant derivation via connection

Let $\pi$ be a fibration and $s:B\to E$ a smooth section, a differentiable map selecting a point $z=s(a)$ in each fiber $F_a$. A connection on the bundle $E$ allows to differentiate $s$ along a smooth path $\gamma:(\R^1,0)\to (B,a)$, with the derivative (at the initial point $a$) being a "vertical" vector tangent to the fiber $F_a$ at the point $z=s(a)\in E$.

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 $$ (\R^1,0)\owns t\mapsto f(t)=\tau_t^0(s(\gamma(t))\in F_a.\tag{TC} $$ If the section and the transport curve $\gamma$ are smooth (the condition that should be verified in the local trivializing charts), then the curve $t\mapsto f(t)$ is a smooth parametrized curve in the fiber $F_a$ which has a well-defined velocity vector $w\in T_z F\subseteq T_z E$, $z=f(0)=s(a)$.

Linearization (computing the differentials) of all maps occurring in (TC) provides the linear relationships between the vectors $u=\dot \gamma(0)\in T_a B$, its image $v=\rd s(a) u$ and its "vertical projection" $w\in T_z F\subset T_z F$, $z=s(a)$. Thus the parallel transport in the tangential (linear) approximation provides us with the splitting of the space $T_z E$, the linear projection $P_z:T_z E\to T_z F$ on the vertical subspace $T_z F$, the kernel of $\rd \pi(z):T_z E\to T_a B$.

The directional derivative of a section $s(\cdot)$ along a curve $\gamma$ tangent to the vector $u=\dot\gamma(a)$ at the point $a$ of the base is therefore the vector $w=P_z\cdot\rd s(a)\cdot\dot\gamma(a)$, with the operators $\{P_z:T_zE\to \operatorname{Ker}\rd\pi(z)\}$ giving the complete infinitesimal description of the connection.

In practice instead of the family of operators $\{P_z\}$ one uses the distribution of their null spaces, a sub-bundle $\Gamma\subset TE$ of the total bundle. Subspaces $\{\Gamma_z\subseteq T_z E\}$ from this distribution are referred to as horizontal subspaces should be complementary to the vertical subspaces $\{V_z=T_z E\}$ tangent to the fibers of the projection $\pi$. This means that $\dim\Gamma=\dim B$ and the differential $\rd \pi$ restricted on these subspaces is invertible.

Ehresmann connection

The Ehresmann connection on the fiber bundle $\pi:E\to B$ is a horizontal distribution $\Gamma\subset TE$ complementary (transversal) to the vertical distribution $V=\operatorname{Ker}\rd \pi\subset TE$.

The Ehresmann connection defines the parallel transport along any smooth path $\gamma:[0,1]\to B$ in the base as follows. For any point $z\in E$ over the curve, $\pi(z)=\gamma(t)$, there is a unique "horizontal lift", the vector $v(z)\in\Gamma_z$, which covers the velocity vector $\dot\gamma(t)$, $\rd \pi(z)v(z)=\dot\gamma(t)$. The corresponding vector field is defined on the induced bundle $\gamma^*\pi$ over the segment $[0,1]$; by construction, the flow of this field commutes with the projection on the base, hence maps fibers to fibers and defines the family of transport maps $\tau_t^s$.

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