Parallel transport

A topological or differential geometric construction generalizing the idea of parallel translation in affine spaces to general bundles, see connection. 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 flat spaces

Some smooth manifolds are naturally equipped with a possibility to freely move tangent vectors from one point to another.

(Genuine) parallel translation in vector and 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 on Lie groups

The idea of parallel transport uses the Lie group structure on $\R^n$ but the commutativity in fact is not necessary.

Let $G$ be a (finite-dimensional) Lie group with $\mathfrak g=T_e G$ the tangent space at the unity. For any element $g\in G$ denote by $r_g:G\to G$ the right action, $r_g(x)=x\cdot g$. This action is transitive and free by the smooth diffeomorphisms. The differential $\rd r_g:T_e G\to T_g G$ is a bijection of the tangent spaces, which allows to identify them. For any ordered tuple of elements $g_1,\dots,g_k\in G$ whose product is equal to $e$, $g_1\cdots g_k=e$, the composition $r_{g_k}\circ\cdots \circ r_{g_1}$ is the identical transformation and the corresponding self-map $T_x G\to T_x G$ the identity map.

The same obviously holds also for the left action $l_g:x\mapsto g\cdot x$ of $G$ on itself and for any free transitive action of $G$ on the corresponding homogeneous space $F$.

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.

Motivation

Although the definition of a bundle requires the bundles to depend "regularly" (continuously or differentiably) on the point in the base, there is no canonical way to identify between themselves points on different fibers.

Example. Consider a smooth surface $B^2$ embedded in the affine space $\R^3$. The tangent space $T_a B$ to the surface at a variable point $a$ depends smoothly on the point, yet there is no obvious way to "translate" a vector tangent to $S$ at one point, to another. However, one can construct such "parallel transport" using the Euclidean structure as follows: if $a_0,a_1$ are two sufficiently close points, then there exists a unique segment $\gamma=[a,b]\subseteq S$ of shortest length connecting them. This segment is smooth and it has well defined velocity vectors $v_i\in T_{a_i}S$. The linear map $\tau_0^1:T_{a_0}S\to T_{a_1}S$ which sends $v_0$ to $v_1$ and is an orientation-preserving isometry, is uniquely defined by these two conditions. This defines the "parallel transport" between two tangent spaces at close points. However, if $a_0,a_1,\dots,a_{n-1}, a_n=a_0$ are $n$ points (even sufficiently close to each other) and $\tau_i^{i+1}:T_{a_i}S\to T_{a_{i+1}}S$ is the corresponding "parallel transport" operator, then the result of the composition $\tau_{n-1}^0\circ \tau_{n-2}^{n-1}\circ\cdots\circ\tau_1^2\circ\tau_0^1$ is in general a nontrivial rotation.

In this article we discuss differential geometric constructions which allow to introduce and study different ways of parallel transport.

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 $F_a$.

Parallel transport for coverings: covering homotopy

If the generc fiber $F$ of a bundle $\pi:E\to B$ has discrete topology, i.e., the only continous maps $[0,1]\to F$ are constants, then there is only one way to choose the point $x(t)\in \pi^{-1}(\gamma(t))$ continuously depending on $t$. In other words, if the fiber $F$ is discrete, the only continuous sections are locally constant.

Thus any covering of a (sufficiently regular) base $B$ admits a natural connection. This connection is locally flat: the result of parallel transport along a sufficiently small closed loop is the identical transformation. In a standard way this implies that the parallel transport between two (distant) fibers depends only on the homotopy class of the path chosen for the parallel transport.

Thus for any fixed point $a\in B$ there is a natural representation of the fundamental group $\pi_1(B,a)$ by bijections of the fiber $F_a\simeq F$, called the group of covering transformations. Change of the base point $a$ results in a conjugate representation (assuming the base $B$ is connected).

Example. Let $\pi:E\to B$ is a fibration of smooth manifolds with a compact fiber $F$. Then any two sufficiently close fibers $F_a$ and $F_b$ are diffeomorphic (not canonically), however, the respective homology groups $H_k(F_a,\Z)$, $k\le \dim F$, considered as elements of a lattice with discrete topology, are canonically isomorphic. This allows to transport cycles of all dimensions on the fibers $F_a$ along any path in the base in a unique way. This connection together with the covariant derivation which it induces on the dual bundle $H^k_{\rd R}(F_a,\R)$ (especially in the algebraic context) (especially in the algebraic context) is called the Gauss-Manin connection.

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 $\tau_t^s$.

Integrability and flatness

The parallel transport between two fibers $F_0=\pi^{-1}(\gamma(0))$ and $F_1=\pi^{-1}(\gamma(1))$ defined by a horizontal distribution $\Gamma$ in general depends on the whole curve $\gamma$ and changes with the curve even if its endpoints remain fixed.

There is one particular case when the parallel transport is (locally) independent of the path. Assume that the horizontal distribution is involutive and by the Frobenius theorem it admits integral submanifolds through each point $z\in E$. By definition, these manifolds are transversal to the fibers and hence locally diffeomorphic to the base: each such manifold is the graph of a locally constant section $s$, whose graph is tangent to the distribution. If the generic fiber is compact, this implies that for any point $a\in B$ on the base, there exists a small neighborhood $U\owns a$ in $B$ such that each fiber $F_b=\pi^{-1}(b)$ is canonically diffeomorphic to $F_a$. The diffeomorphism maps every point $z\in F_a$ to the value $s_z(b)\in F_b$, where $s_z(\cdot)$ is the uniquely defined horizontal section of $\pi$ such that $s_z(a)=z$. In particular, the parallel transport of the fiber $F_a$ on itself is identical for any closed loop with $\gamma(0)=\gamma(1)=a$ (smooth or only continuous) which entirely belongs to $U$, $\gamma([0,1])\subset U$.

Connections with the property that the parallel translation along any sufficiently small (contractible) loop on the base is identical, are called (locally) flat connections. The local flatness, however, does not guarantee that the parallel transport along a "large" (non-contractible) loop in the base will be identical. Ditto for singular connections exhibiting singularities on small subsets of the base.

Holonomy

For a fixed base point $a\in B$ and the corresponding fiber, one may consider all self-maps of $F_a$ which appear as the result of parallel transport along closed curves with the beginning and end at $a$. The result clearly has the structure of a topological group, which in many interesting cases turns out to be finite-dimensional (eventually, discrete) Lie group. This group is called the holonomy group of the connection.

Connections on principal and vector bundles

Non-involutive distributions produce parallel transport that is non-identical even for arbitrarily small simple closed loops: the deviation from the identity is generally referred to as the curvature of the connection.

$\def\e{\varepsilon}$ To separate the contribution of different elements of the construction, consider two non-collinear vectors $u_1,u_2\in T_a B$ tangent to the base, and a small loop $\gamma_\e$ which (say, in a given chart $U$ on the base) corresponds to oriented boundary the parallelogram with the edges $\e u_1$ and $\e u_2$ for a sufficiently small real number $\e>0$. Although the loop $\gamma_\e$ is only piecewise-smooth, one can attempt to compute the corresponding parallel transport map $\tau_\e=\tau_{\gamma_\e}:F_a\to F_a$. An easy estimate made in the trivializing coordinates shows that if the generic fiber $F$ is compact, then $\tau_\e$ is a self-diffeomorphism of the fiber, $O(\e^2)$-close to the identity.

In absence of another structure on the generic fiber $F$ it is difficult to say more, in particular to study how the properly normalized displacement $\lim_{\e\to0}\e^{-2}(\tau_\e(x)-x)$ depends on the point $x\in F_a$ and the two vectors $u_1,u_2$. However, for the special cases where the generic fiber is "flat" (a nicely parallelizable homogeneous space, see above, e.g., for vector or principal G-bundles) the computation leads to the notion of the curvature tensor, resp., curvature form.

The notions and constructions which are formulated in terms of connections and parallel transport, are central for many areas of geometry and mathematical physics. We mention only few keywords, referring to the corresponding articles for more information.

• An affine connection on the tangent bundle $TM$ defines the class of "auto-parallel curves" on $M$, called geodesic lines. The tangent vector of a geodesic curve remains parallel to itself along these curves. The dynamical properties of the respective geodesic flows are radically different for manifolds of positive and negative curvature.
• The tangent bundle of a Riemannian manifold admits a unique Levi-Civita connection which generates isometric parallel transport. Geodesic curves if this connection are local extremals for the Riemannian length functional. Similar "distinguished" connections exist also for Kähler manifolds.
• Using connections, one can define certain characteristic classes, which measure various aspects of the nontriviality of the vector bundles on which they are defined.

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