- Parallel transport
In

geometry ,**parallel transport**is a way of transporting geometrical data along smooth curves in amanifold . If the manifold is equipped with anaffine connection (acovariant derivative or connection on thetangent bundle ), then this connection allows one to transport vectors of the manifold along curves so that they stay "parallel" with respect to the connection. Other notions of connection come equipped with their own parallel transportation systems as well. For instance, aKoszul connection in avector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann orCartan connection supplies a "lifting of curves" from the manifold to the total space of aprincipal bundle . Such curve lifting may sometimes be thought of as the parallel transport ofreference frame s.The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of "connecting" the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a "connection". In fact, the usual notion of connection is the

infinitesimal analog of parallel transport. Or, "vice versa", parallel transport is the local realization of a connection.As parallel transport supplies a local realization of the connection, it also supplies a local realization of the

curvature known asholonomy . TheAmbrose-Singer theorem makes explicit this relationship between curvature and holonomy.**Parallel transport on a vector bundle**Let "M" be a smooth manifold. Let "E"→"M" be a vector bundle with

covariant derivative ∇ and "γ": "I"→"M" a smooth curve parameterized by an open interval "I". A section $X$ of $E$ along "γ" is called**parallel**if:$abla\_\{dotgamma(t)\}X=0\; ext\{\; for\; \}t\; in\; I.,$

Suppose we are given an element "e"

_{0}∈ "E"_{"P"}at "P" = "γ"(0) ∈ "M", rather than a section. The**parallel transport**of "e"_{0}along "γ" is the extension of "e"_{0}to a parallel "section" "X" on "γ".More precisely, "X" is the unique section of "E" along "γ" such that

#$abla\_\{dot\{gamma\; X\; =\; 0$

#$X\_\{gamma(0)\}\; =\; e\_0.$Note that in alocal trivialization (1) defines anordinary differential equation , with theinitial condition given by (2). Thus thePicard–Lindelöf theorem guarantees the existence and uniqueness of the solution.Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides

linear isomorphism s between the fibers at points along the curve::$Gamma(gamma)\_s^t\; :\; E\_\{gamma(s)\}\; ightarrow\; E\_\{gamma(t)\}$from the vector space lying over γ("s") to that over γ("t"). This isomorphism is known as the**parallel transport**map associated to the curve. The isomorphisms between fibers obtained in this way will in general depend on the choice of the curve: if they do not then parallel transport along every curve can be used to define parallel sections of "E" over all of "M". This is only possible if the**curvature**of ∇ is zero.In particular, parallel transport around a closed curve starting at a point "x" defines an

automorphism of the tangent space at "x" which is not necessarily trivial. The parallel transport automorphisms defined by all closed curves based at "x" form a transformation group called theholonomy group of ∇ at "x". There is a close relation between this group and the value of the curvature of ∇ at "x"; this is the content of theAmbrose-Singer holonomy theorem .**Recovering the connection from the parallel transport**Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition $scriptstyle\{\; abla\_\{dot\{gamma=0\}$. Conversely, if a suitable notion of parallel transport is available, then a corresponding connection can be obtained by differentiation. This approach is due, essentially, to harvtxt|Knebelman|1951; see harvtxt|Guggenheimer|1977. harvtxt|Lumiste|2001 also adopts this approach.

Consider an assignment to each curve γ in the manifold a collection of mappings:$Gamma(gamma)\_s^t\; :\; E\_\{gamma(s)\}\; ightarrow\; E\_\{gamma(t)\}$such that

# $Gamma(gamma)\_s^s\; =\; Id$, the identity transformation of "E"_{γ(s)}.

# $Gamma(gamma)\_u^tcircGamma(gamma)\_s^u\; =\; Gamma(gamma)\_s^t.$

# The dependence of Γ on γ, "s", and "t" is "smooth."The notion of smoothness in condition 3. is somewhat difficult to pin down (see the discussion below of parallel transport in fibre bundles). In particular, modern authors such as Kobayashi and Nomizu generally view the parallel transport of the connection as coming from a connection in some other sense, where smoothness is more easily expressed.Nevertheless, given such a rule for parallel transport, it is possible to recover the associated infinitesimal connection in "E" as follows. Let γ be a differentiable curve in "M" with initial point γ(0) and initial tangent vector "X" = γ′(0). If "V" is a section of "E" over γ, then let:$abla\_X\; V\; =\; lim\_\{h\; o\; 0\}frac\{Gamma(gamma)\_h^0V\_\{gamma(h)\}\; -\; V\_\{gamma(0)\{h\}\; =\; left.frac\{d\}\{dt\}Gamma(gamma)\_t^0V\_\{gamma(t)\}\; ight|\_\{t=0\}.$This defines the associated infinitesimal connection ∇ on "E". One recovers the same parallel transport Γ from this infinitesimal connection.

**pecial case: The tangent bundle**Let "M" be a smooth manifold. Then a connection on the

tangent bundle of "M", called anaffine connection , distinguishes a class of curves called (affine)geodesic s harv|Kobayashi|Nomizu|loc=Volume 1, Chapter III. A smooth curve "γ": "I" → "M" is an**affine geodesic**if $dotgamma$ is parallel transported along $gamma$, that is:$Gamma(gamma)\_s^tdotgamma(s)\; =\; dotgamma(t).,$

Taking the derivative with respect to time, this takes the more familiar form

:$abla\_\{dotgamma(t)\}dotgamma\; =\; 0.,$

**Parallel transport in Riemannian geometry**In (pseudo)

Riemannian geometry , ametric connection is any connection whose parallel transport mappings preserve themetric tensor . Thus a metric connection is any connection Γ such that, for any two vectors "X", "Y" ∈ T_{γ(s)}:$langleGamma(gamma)\_s^tX,Gamma(gamma)\_s^tY\; angle\_\{gamma(t)\}=langle\; X,Y\; angle\_\{gamma(s)\}.$Taking the derivative at "t"=0, the associated differential operator ∇ must satisfy a product rule with respect to the metric::$abla\_Zlangle\; X,Y\; angle\; =\; langle\; abla\_ZX,Y\; angle\; +\; langle\; X,\; abla\_Z\; Y\; angle.$

**Geodesics**If ∇ is a metric connection, then the affine geodesic are the usual

geodesic s of Riemannian geometry and are the locally distance minimizing curves. More precisely, first note that if "γ": "I" → "M", where "I" is an open interval, is a geodesic, then the norm of $dotgamma$ is constant on "I". Indeed:$frac\{d\}\{dt\}langledotgamma(t),dotgamma(t)\; angleigg|\_\{t=0\}\; =\; 2langle\; abla\_\{dotgamma(t)\}dotgamma(t),dotgamma(t)\; angle=0.$It follows that if "A" is the norm of $dotgamma(t)$ then the distance, induced by the metric, between two "close enough" points on the curve "γ", say "γ"("t"

_{1}) and "γ"("t"_{2}), is given by:$mbox\{dist\}ig(gamma(t\_1),gamma(t\_2)ig)\; =\; A|t\_1\; -\; t\_2|.$The formula above might not be true for points which are not close enough since the geodesic might for example wrap around the manifold (e.g. on a sphere).**Generalizations**The parallel transport can be defined in greater generality for other types of connections, not just those defined in a vector bundle. One generalization is for principal connections harv|Kobayashi|Nomizu|1996|loc=Volume 1, Chapter II. Let "P" → "M" be a

principal bundle over a manifold "M" with structureLie group "G" and a principal connection ω. As in the case of vector bundles, a principal connection ω on "P" defines, for each curve γ in "M", a mapping:$Gamma(gamma)\_s^t\; :\; P\_\{gamma(s)\}\; ightarrow\; P\_\{gamma(t)\}$from the fibre over γ("s") to that over γ("t"), which is an isomorphism ofhomogeneous space s: i.e. $Gamma\_\{gamma(s)\}\; gu\; =\; gGamma\_\{gamma(s)\}$ for each "g"∈"G".Further generalizations of parallel transport are also possible. In the context of

Ehresmann connection s, where the connection depends on a special notion of "horizontal lifting" of tangent spaces, one can define parallel transport via horizontal lifts.Cartan connection s are Ehresmann connections with additional structure which allows the parallel transport to be though of as a map "rolling" a certain model space along a curve in the manifold. This rolling is called development.**ee also***

Basic introduction to the mathematics of curved spacetime

*Connection (mathematics)

*Development (differential geometry)

*Affine connection

*Covariant derivative

*Geodesic (general relativity)

*Lie derivative **References***

*citation | author=Knebelman|title=Spaces of relative parallelism|journal=Annals of Mathematics|volume=53|series=2|pages=387-399|year=1951|url=http://links.jstor.org/sici?sici=0003-486X(195105)2%3A53%3A3%3C387%3ASORP%3E2.0.CO%3B2-6

*citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry, Volume 1| publisher=Wiley-Interscience | year=1996|id = ISBN 0471157333; Volume 2, ISBN 0471157325.

*springer|id=c/c025180|title=Connections on a manifold|first=Ü.|last=Lumiste|year=2001**External links*** [

*http://torus.math.uiuc.edu/jms/java/dragsphere/ Spherical Geometry Demo*] . An applet demonstrating parallel transport of tangent vectors on a sphere.

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