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A very flexible construction aimed to represent a family of similar objects (fibres or fibers, depending on the preferred spelling) which are parametrized by the index set which itself has an additional structure (topological space, smooth manifold etc.).
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{{TEX|done}}
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{{MSC|53C05}}
  
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The most known examples are the tangent and cotangent bundles of a smooth manifold. The coverings are also a special particular form of a topological bundle (with discrete fibers).  
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A topological or differential geometric construction generalizing the idea of [[parallel translation]] in affine spaces to fibers of 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]].  
  
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==Formal definition of a topological bundle==
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==Parallel transport (translation) in flat spaces==
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Let $\pi:E\to B$ be a continuous map between topological spaces, called the ''total space''<ref>Also names [[fibre space]] or ''fibered space'' are used. </ref> and the ''base'', and $F$ yet another topological space called ''fiber'', such that the preimage $F_b=\pi^{-1}(b)\subset E$ of every point of the base is [[homeomorphism|homeomorphic]] to $X$. The latter condition means that $E$ is the disjoint union of "fibers", $E=\bigsqcup_{b\in B} F_b$ homeomorphic to each other.
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Some smooth manifolds are naturally equipped with a possibility to freely move tangent vectors from one point to another.
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===(Genuine) parallel translation in vector and affine spaces===
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The map $\pi$ is called [[fibration]]<ref>Also the terms ''bundle'' or ''fiber bundle'' are used.</ref> of $E$ over $B$, if the above representation is ''locally trivial'': any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is [[topological equivalence|topologically equivalent]] to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$.
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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 $\{t_w:w\in V\}$:   
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====Examples====
 
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# The trivial bundle $E=F\times B$,  $\pi=\pi_2: F\times B\to B$, $(v,b)\mapsto b$. In this case all trivializing homeomorphisms are globally defined on the entire total space (as the identity map).
 
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# Let $E=\R^n\smallsetminus\{0\}$ be the punctured Euclidean space, $B=\mathbb S^{n-1}$ the standard unit sphere and $\pi$ the radial projection $\pi(x)=\|x\|^{-1}\cdot x$. This is a topological bundle with the fiber $F=(0,+\infty)\simeq\R^1$.
 
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# Let $E=\mathbb S^{n-1}$ as above, $B=\R P^{n-1}$ the real [[projective space]] (all lines in $\R^n$ passing through the origin) and $\pi$ the map taking a point $x$ on the sphere into the line $\ell_x$ passing through $x$. The preimage $\pi^{-1}(\ell)$ consists of two antipodal points $x$ and $-x\in\mathbb S^{n-1}$, thus $F$ is a discrete two-point set $\mathbb Z_2=\{-1,1\}$. This is a topological bundle, which cannot be trivial: indeed, if it were, then the total space $\mathbb S^{n-1}$ would consist of two connected components, while it is connected.
 
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# More generally, let $\pi:M^m\to N^n$ be a differentiable map between two smooth (connected) compact manifolds of dimensions $m\ge n$, which has the maximal rank (equal to $n$) everywhere. One can show then, using the implicit function theorem and partition of unity, that $\pi$ is a topological bundle with a fiber $F$ which itself is a smooth compact manifold<ref>This statement is also known as the Ehresmann theorem, see Ehresmann, C., ''Les connexions infinitésimales dans un espace fibré différentiable'', Colloque de Topologie, Bruxelles (1950), 29-55. The compactness assumption can be relaxed by the requirement that the map $\pi$ is ''proper'', i.e., preimage of any compact in $N$ is a compact in $M$. </ref>
 
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===Cocycle of a bundle===
 
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On a nonvoid overlapping $U_{\alpha\beta}=U_\alpha\cap U_\beta$ of two different trivializing charts $U_\alpha$ and $U_\beta$ two homeomorphisms $H_\alpha,H_\beta: \pi^{-1}(U_{\alpha\beta})\to F\times U_{\alpha\beta}$ are defined. Since both $H_\alpha$ and $H_\beta$ conjugate $\pi$ with the Cartesian projection on $U_{\alpha\beta}$, they map each fiber $F_b=\pi^{-1}(b)$ into the same space $F\times\{b\}$. The composition $H_\alpha\circ H_\beta^{-1}$ keeps constant the $b$-component and hence takes the "triangular" form
 
 
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H_\alpha\circ H_\beta^{-1}:(v,b)\mapsto (H_{\alpha\beta}(b,v),b),\qquad H_{\alpha\beta}(\cdot,b)\in\operatorname{Homeo}(F)
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\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).
 
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with the homeomorphisms $H_{\alpha\beta}(\cdot, b)$ continuously depending on $b\in U_{\alpha\beta}$. The collection of these "homeomorphism-valued" functions defined in the intersections $U_{\alpha\beta}$ is called the [[cocycle]] associated with a given trivialization of the bundle $\pi$ (or simply the ''cocycle of the bundle''. They homeomorphisms $\{H_{\alpha\beta}\}$ satisfy the following identities, obvious from their construction:
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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]],
 
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H_{\alpha\beta}\circ H_{\beta\alpha}=\operatorname{id},\qquad H_{\alpha\beta}\circ H_{\beta\gamma}\circ H_{\gamma\alpha}=\operatorname{id},
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\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.
 
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the second being true on every nonvoid triple intersection $U_{\alpha\beta\gamma}=U_\alpha\cap U_\beta\cap U_\gamma$.
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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 $a$.
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These trivial observations indicate some of the properties that will fail for general parallel transport.
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{{anchor|flat}}
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===Parallel transport on Lie groups===
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The idea of parallel transport uses the Lie group structure on $\R^n$ but the commutativity in fact is not necessary.
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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 of a group on a manifold|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.
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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]] $F$.
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==Parallel transport in topological bundles and fibrations==
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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]].
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===Motivation===
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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.
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'''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.
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In this article we discuss differential geometric constructions which allow to introduce and study different ways of parallel transport.
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===Formal definition===
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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:
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* $\tau_t^s\circ \tau_r^t=\tau_{r}^s$ for all values $r,t,s\in[0,1]$ in any order, $\tau_t^t\equiv\operatorname{id}$,
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* the homeomorphisms $\tau_{t}^s$ continuously depend on $t,s\in[0,1]$,
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* the homemorphisms $\tau_t^s$ preserve the additional structure<ref>E.g., if all fibers are linear or Euclidean spaces, then $\tau_t^s$ must be linear, resp., linear orthogonal operators.</ref> on the fibers, if any.
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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 $F_a$.
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===Parallel transport for coverings: covering homotopy===
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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<ref>Assuming the base $B$ not very pathological, e.g., locally simply connected.</ref> 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<ref>I.e., $s(a)=\operatorname{const}$ in any trivializing chart.</ref>.
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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.
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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 transformation]]s. Change of the base point $a$ results in a conjugate representation (assuming the base $B$ is connected).
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'''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 group]]s $H_k(F_a,\Z)$, $k\le \dim F$, 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]].
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===Smooth connections on fibrations===
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''Main article: [[Connection]]''.
  
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===Vector bundles and other additional structures on the fibers===
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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]].
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The general construction of bundle easily allows various additional structures, both on the base space and (more importantly) on the fibers. By far the most important special case is that of [[vector bundle]]s.
 
  
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To define a vector bundle, one has in addition to the principal definition assume the following:
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====Covariant derivation via connection====
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# The fiber $F$ is a vector space<ref>The fiber $F$ should be equipped with some topology, but often it is finite-dimensional, $F\simeq\R^n$ or $F\simeq\C^n$, thus leaving only the default option.</ref>, and
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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 $z=s(a)\in E$.
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# The trivializing homeomorphisms must respect the linear structure of the fibers.  
 
  
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The second assumption means that rather than being arbitrary homeomorphisms, the maps $\{H_{\alpha\beta}\}$ forming the bundle cocycle, must be linear invertible of each "standard fiber" $F\times \{b\}$; if the fiber is identified with the canonical $n$-space $\Bbbk^n$ (over $\Bbbk=\R$ or $\Bbbk=\C$), then the cocycle will consist of ''invertible continuous matrix-functios'' $M_{\alpha\beta}:U_{\alpha\beta}\to\operatorname{GL}(n,\Bbbk)$, so that $H_{\alpha\beta}(v,b)=(M_{\alpha\beta}(b)\, v, b)$, $v\in\Bbbk^n$. The cocycle identities become then the identites relating the values of these matrix-valued functions,
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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
 
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M_{\alpha\beta}(b)\cdot M_{\beta\alpha}(b)\equiv E,\qquad M_{\alpha\beta}(b)\cdot M_{\beta\gamma}(b)\cdot M_{\gamma\alpha}(b)\equiv E,
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(\R^1,0)\owns t\mapsto f(t)=\tau_t^0(s(\gamma(t))\in F_a.\tag{TC}
 
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where $E$ is the $n\times n$-identical matrix.
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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 $w\in T_z F\subseteq T_z E$, $z=f(0)=s(a)$. 
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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<ref>It should be stressed that the linear projection $P_z$ of a tangent space onto its subspace is not a differential of any suitable smooth map.</ref> $P_z:T_z E\to T_z F$ on the vertical subspace $T_z F$, the kernel of $\rd \pi(z):T_z E\to T_a B$.
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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.
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In practice instead of the family of operators $\{P_z\}$ one uses the [[distribution of tangent subspaces|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 distribution|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.
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===={{anchor|ehresmann}}Ehresmann connection====
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The Ehresmann connection on the fiber bundle $\pi:E\to B$ is a [[horizontal distribution]] $\Gamma\subset TE$ complementary (transversal) to the ''vertical''<ref>Note that the vertical distribution is [[integrability|integrable]] ([[involutive distribution|involutive]]), while the integrability of $\Gamma$ is not postulated.</ref> distribution $V=\operatorname{Ker}\rd \pi\subset TE$.
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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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For vector bundles all linear constructions become well defined on fibers.
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====Integrability and flatness====
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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.  
  
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Following the way, one may define vector bundles with extra algebraic structures on the fibers. For instance, if the cocycle defining the bundle, consists of orthogonal matrices, $M_{\alpha\beta}:U_{\alpha\beta}\to\operatorname{SO}(n,\R)$, then the fibers of the bundle naturally acquire the structure of Euclidean spaces. Other natural examples are bundles whose fibers have the [[Hermitian structure]] (the cocycle should consist of unitary matrix functions then) or [[symplectic structure|symplectic]] spaces (with canonical cocycle matrices preserving the symplectic structure).  
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There is one particular case when the parallel transport is (locally) independent of the path. Assume that the horizontal distribution is [[involutive distribution|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 $U$, $\gamma([0,1])\subset U$.
  
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====Tangent and cotangent bundle of a smooth manifold====
 
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If $M$ is a smooth manifold with the atlas of coordinate charts $\{U_\alpha\}$ and the maps $h_\alpha:U_\alpha\to\R^m$, then the differentials of these maps $\rd h_\alpha$ allow to identify the tangent space $T_a M$ at $a\in U_{\alpha}$ with $\R^m$ and the union $\bigsqcup_{a\in U_\alpha}T_a M$ with $\R^m\times U_\alpha$ (we write the tangent vector first). For a point $a\in U_{\alpha\beta}$ there are two identifications which differ by the Jacobian matrix of the transition map $h_{\alpha\beta}=h_\alpha\circ h_\beta^{-1}$. This shows that the tangent bundle $TM$ is indeed a vector bundle in the sense of the above definition.
 
  
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The cotangent bundle is also trivialized by every atlas $\{h_\alpha:U_\alpha\to\R^m\}$ on $M$, yet in this case the direction of arrows should be reverted<ref>Covectors form a covariant rather than contravariant tensor of rank $1$.</ref>: the cotangent space $T_a^*M$ is identified with $\R^n$ by the linear map $(\rd h_\alpha^*)$, thus the corresponding cocycle will consist of the transposed inverse Jacobian matrices.  
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Connections with the property that the parallel translation along any sufficiently small (contractible) loop on the base is identical, are called (locally) [[flat connection]]s. 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.
  
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<small>
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====Holonomy====
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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.
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==Connections on principal and vector bundles==
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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.
  
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====Equivalence of cocycles====
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$\def\e{\varepsilon}$
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The trivializing maps defining the structure of a bundle (vector or topological) are by no means unique, even if the covering domains $U_\alpha$ remain the same. E.g., one can replace the the collection of maps $\{H_{\alpha}\}$ trivializing a vector bundle, by another collection $\{H'_{\alpha}\}$, post-composing them with the maps $F\times U_\alpha\to F\times U_\alpha$, $(v,b)\mapsto (C_\alpha(b)\,v, b)$ with invertible continuous matrix functions $C_\alpha:U_\alpha\to\operatorname{GL}(n,\Bbbk)$. The corresponding matrix cocycle $\{M_{\alpha\beta}\}$ will be replaced then by the new matrix cocycle $\{M'_{\alpha\beta}(b)\}$,
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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.
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$$
 
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M'_{\alpha\beta}(b)=C_\alpha(b)M_{\alpha\beta}(b)C_\beta^{-1}(b),\qquad b\in U_{\alpha\beta}.
 
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\tag{CE}
 
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$$
 
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Two matrix cocycles related by these identities, are called ''equivalent'' and clearly define the same bundle.  
 
  
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'''Example'''. The trivial cocycle $\{M_{\alpha\beta}(b)\}=\{E\}$ which consists of identity matrices, corresponds to the trival bundle $F\times B$: the trivializing maps agree with each other on the intersections and hence define the global trivializing map $H:E\to F\times B$. A cocycle equal to the trivial one in the sense of (CE) is called ''solvable'': its solution is a collection of invertible matrix functions $C_\alpha:U_\alpha\to\operatorname{GL}(n,\Bbbk)$ such that on the overlapping of the domains $U_{\alpha\beta}=U_\alpha\cap U_\beta$ the identities
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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 $u_1,u_2$ (see [[Non-linear connection]]). However, for the special cases where the generic fiber is "flat" (a nicely parallelizable homogeneous space, see [[#flat|above]], e.g., for vector or principal G-bundles) the computation leads to the notion of the [[curvature tensor]], resp., [[curvature form]].
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$$
 
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M_{\alpha\beta}(b)=C_{\alpha}^{-1}(b)C_\beta(b),\qquad \forall\alpha,\beta,\ b\in U_{\alpha}\cap U_\beta.
 
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$$
 
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Thus solvability of cocycle is an analytic equivalent of the topological triviality of the bundle.  
 
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</small>
 
  
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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.
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* An affine connection on the tangent bundle $TM$ defines the class of "auto-parallel curves" on $M$, called [[geodesic line]]s. The tangent vector of a geodesic curve remains parallel to itself along these curves. The dynamical properties of the respective [[geodesic flow]]s are radically different for manifolds of positive and negative curvature.
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* 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 manifold]]s.
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* Using connections, one can define certain [[characteristic class]]es, which measure various aspects of the nontriviality of the vector bundles on which they are defined.
  
  
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<small>
 
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<small>
 
 
<references/>
 
<references/>
 
</small>
 
</small>
  
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==Special classes of bundles==
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==References==
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Together with vector bundles, there are other special classes of bundles.
 
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===$G$-bundles and principal bundles===
 
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Assume that the fiber $F$ of a bundle $\pi:E\to B$ has a structure of the topological group (of the Lie group) $G$ and this group acts on each fiber, say, by the ''right'' multiplication, generating thus the action of $G$ on the total space $E$. Then this action should be consistent with the local trivializations $H_\alpha:\pi^{-1}(U_\alpha)\to G\times U_\alpha$: the corresponding transition maps $H_{\alpha\beta}(\cdot,b):G\to G$ must commute with the right action of $G$. This means that
 
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$$
 
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\forall g\in G, \ b\in B\qquad H_{\alpha\beta}(g,b)=H_{\alpha\beta}(e\cdot g,b)=H_{\alpha\beta}(e,b)\cdot g =g_{\alpha\beta}(b)\cdot g,
 
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\tag{T}
 
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$$
 
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where $g_{\alpha\beta}=H_{\alpha\beta}(e)\in G$ is the uniquely defined group element (depending continuously on $b\in B$), and $e\in G$ is the unit of the group. Thus the $G$-bundle is completely determined by the cocycle $\{g_{\alpha\beta}:U_{\alpha\beta}\to G\}$ satisfying the cocycle identites,
 
−
$$
 
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  g_{\alpha\beta}(\cdot)g_{\beta\alpha}(\cdot)\equiv e,\qquad g_{\alpha\beta}(\cdot)g_{\beta\gamma}(\cdot)g_{\gamma\alpha}(\cdot)\equiv e.
 
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\tag{GC}
 
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$$
 
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Such a bundle (defined by the ''left'' $G$-action of multiplication by $g_{\alpha\beta}$ in the transition maps (T)) is called a [[principal fibre bundle|principal $G$-bundle]].
 
  
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This construction allows to associate (tautologically) with each vector bundle $\pi:E\to B$ with a fiber $\Bbbk^n$ a principal $G$-bundle $\Pi:\mathbf E\to B$ with the same base, where $G=\operatorname{GL}(n,\Bbbk)$ or its proper (matrix) subgroup. Analytically this is achieved by considering the same matrix cocycle $\{M_{\alpha\beta}(\cdot)\}$ and re-interpreting it as the $G$-valued cocycle (GC), $g_{\alpha\beta}=M_{\alpha\beta}$. This bundle is (not surprisingly) called the ''associated principal bundle''.
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See the [[Bundle#literature|recommended literature]] for [[bundle]]s.

Latest revision as of 08:44, 12 December 2013

2020 Mathematics Subject Classification: Primary: 53C05 [MSN][ZBL]

A topological or differential geometric construction generalizing the idea of parallel translation in affine spaces to fibers of 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[1] 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[2] 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[3].

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

Main article: Connection.

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[4] $P_z:T_z E\to T_z F$ 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[5] 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 $u_1,u_2$ (see Non-linear connection). 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.


  1. ↑ E.g., if all fibers are linear or Euclidean spaces, then $\tau_t^s$ must be linear, resp., linear orthogonal operators.
  2. ↑ Assuming the base $B$ not very pathological, e.g., locally simply connected.
  3. ↑ I.e., $s(a)=\operatorname{const}$ in any trivializing chart.
  4. ↑ It should be stressed that the linear projection $P_z$ of a tangent space onto its subspace is not a differential of any suitable smooth map.
  5. ↑ Note that the vertical distribution is integrable (involutive), while the integrability of $\Gamma$ is not postulated.

References

See the recommended literature for bundles.

How to Cite This Entry:
Parallel transport. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_transport&oldid=26185
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