Baker–Campbell–Hausdorff formula: Difference between revisions
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[[Image:Parallel transport.png|thumb|Parallel transport of a vector around a closed loop (from A to N to B and back to A) on the sphere. The angle by which it twists, <math>\alpha</math>, is proportional to the area inside the loop.]] | |||
In [[geometry]], '''parallel transport''' is a way of transporting geometrical data along smooth curves in a [[manifold]]. If the manifold is equipped with an [[affine connection]] (a [[covariant derivative]] or [[connection (vector bundle)|connection]] on the [[tangent 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 (mathematics)|connection]] come equipped with their own parallel transportation systems as well. For instance, a [[Koszul connection]] in a [[vector bundle]] also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An [[Ehresmann connection|Ehresmann]] or [[Cartan connection]] supplies a ''lifting of curves'' from the manifold to the total space of a [[principal bundle]]. Such curve lifting may sometimes be thought of as the parallel transport of [[frame of reference|reference frames]]. | |||
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 as [[holonomy]]. The [[Ambrose-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 [[curve|smooth curve]] parameterized by an open interval ''I''. A [[Section (fiber bundle)|section]] <math>X</math> of <math>E</math> along ''γ'' is called '''parallel''' if | |||
:<math>\nabla_{\dot\gamma(t)}X=0\text{ for }t \in I.\,</math> | |||
Suppose we are given an element ''e''<sub>0</sub> ∈ ''E''<sub>''P''</sub> at ''P'' = ''γ''(0) ∈ ''M'', rather than a section. The '''parallel transport''' of ''e''<sub>0</sub> along ''γ'' is the extension of ''e''<sub>0</sub> to a parallel ''section'' ''X'' on ''γ''. | |||
More precisely, ''X'' is the unique section of ''E'' along ''γ'' such that | |||
#<math>\nabla_{\dot\gamma} X = 0 </math> | |||
#<math>X_{\gamma(0)} = e_0.</math> | |||
Note that in any given coordinate patch, (1) defines an [[ordinary differential equation]], with the [[initial condition]] given by (2). Thus the [[Picard–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: | |||
:<math>\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}</math> | |||
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 '''[[connection (vector bundle)|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 the [[holonomy 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 the [[Ambrose-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 <math>\scriptstyle{\nabla_{\dot{\gamma}}=0}</math>. 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 | |||
:<math>\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}</math> | |||
such that | |||
# <math>\Gamma(\gamma)_s^s = Id</math>, the identity transformation of ''E''<sub>γ(s)</sub>. | |||
# <math>\Gamma(\gamma)_u^t\circ\Gamma(\gamma)_s^u = \Gamma(\gamma)_s^t.</math> | |||
# 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 | |||
:<math>\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \left.\frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\right|_{t=0}.</math> | |||
This defines the associated infinitesimal connection ∇ on ''E''. One recovers the same parallel transport Γ from this infinitesimal connection. | |||
===Special case: The tangent bundle=== | |||
Let ''M'' be a smooth manifold. Then a connection on the [[tangent bundle]] of ''M'', called an [[affine 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 <math>\dot\gamma</math> is parallel transported along <math>\gamma</math>, that is | |||
:<math>\Gamma(\gamma)_s^t\dot\gamma(s) = \dot\gamma(t).\,</math> | |||
Taking the derivative with respect to time, this takes the more familiar form | |||
:<math>\nabla_{\dot\gamma(t)}\dot\gamma = 0.\,</math> | |||
==Parallel transport in Riemannian geometry== | |||
In ([[pseudo-Riemannian manifold|pseudo]]) [[Riemannian geometry]], a [[metric connection]] is any connection whose parallel transport mappings preserve the [[metric tensor]]. Thus a metric connection is any connection Γ such that, for any two vectors ''X'', ''Y'' ∈ T<sub>γ(s)</sub> | |||
:<math>\langle\Gamma(\gamma)_s^tX,\Gamma(\gamma)_s^tY\rangle_{\gamma(t)}=\langle X,Y\rangle_{\gamma(s)}.</math> | |||
Taking the derivative at ''t''=0, the associated differential operator ∇ must satisfy a product rule with respect to the metric: | |||
:<math>Z\langle X,Y\rangle = \langle \nabla_ZX,Y\rangle + \langle X,\nabla_Z Y\rangle.</math> | |||
===Geodesics=== | |||
If ∇ is a metric connection, then the affine geodesics 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 <math>\dot\gamma</math> is constant on ''I''. Indeed | |||
:<math>\frac{d}{dt}\langle\dot\gamma(t),\dot\gamma(t)\rangle = 2\langle\nabla_{\dot\gamma(t)}\dot\gamma(t),\dot\gamma(t)\rangle =0.</math> | |||
It follows from an application of [[Gauss's_lemma_(Riemannian_geometry)|Gauss's Lemma]] that if ''A'' is the norm of <math>\dot\gamma(t)</math> then the distance, induced by the metric, between two ''close enough'' points on the curve ''γ'', say ''γ''(''t''<sub>1</sub>) and ''γ''(''t''<sub>2</sub>), is given by | |||
:<math>\mbox{dist}\big(\gamma(t_1),\gamma(t_2)\big) = A|t_1 - t_2|.</math> | |||
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 [[connection (principal bundle)|principal connections]] {{harv|Kobayashi|Nomizu|1996|loc=Volume 1, Chapter II}}. Let ''P'' → ''M'' be a [[principal bundle]] over a manifold ''M'' with structure [[Lie 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 | |||
:<math>\Gamma(\gamma)_s^t : P_{\gamma(s)} \rightarrow P_{\gamma(t)}</math> | |||
from the fibre over γ(''s'') to that over γ(''t''), which is an isomorphism of [[homogeneous space]]s: i.e. <math>\Gamma_{\gamma(s)} gu = g\Gamma_{\gamma(s)}</math> 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 space|horizontal lift]]ing" of tangent spaces, one can define [[Ehresmann_connection#Parallel_transport_via_horizontal_lifts|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 [[Klein geometry|model space]] along a curve in the manifold. This rolling is called [[development (differential geometry)|development]]. | |||
==Approximation: Schild's ladder== | |||
[[Image:Schild's ladder step 4.svg|thumb|right|Two rungs of [[Schild's ladder]]. The segments ''A''<sub>1</sub>''X''<sub>1</sub> and ''A''<sub>2</sub>''X''<sub>2</sub> are an approximation to first order of the parallel transport of ''A''<sub>0</sub>''X''<sub>0</sub> along the curve.]] | |||
{{main|Schild's ladder}} | |||
Parallel transport can be discretely approximated by [[Schild's ladder]], | |||
which takes finite steps along a curve, and approximates | |||
[[Levi-Civita parallelogramoid]]s by approximate [[parallelogram]]s. | |||
==See also== | |||
* [[Basic introduction to the mathematics of curved spacetime]] | |||
* [[Connection (mathematics)]] | |||
* [[Development (differential geometry)]] | |||
* [[Affine connection]] | |||
* [[Covariant derivative]] | |||
* [[Geodesic (general relativity)]] | |||
* [[Lie derivative]] | |||
* [[Schild's ladder]] | |||
* [[Levi-Civita parallelogramoid]] | |||
==References== | |||
* {{citation|first=Heinrich|last=Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|isbn=0-486-63433-7}} | |||
*{{citation | author=Knebelman|title=Spaces of relative parallelism|journal=Annals of Mathematics|volume=53|series=2|pages=387–399|year=1951| doi=10.2307/1969562 | issue=3 | publisher=The Annals of Mathematics, Vol. 53, No. 3 | jstor=1969562}} | |||
*{{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]], Volume 1| publisher=Wiley-Interscience | year=1996 | isbn=0-471-15733-3}}; Volume 2, ISBN 0-471-15732-5. | |||
*{{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. | |||
[[Category:Riemannian geometry]] | |||
[[Category:Connection (mathematics)]] |
Revision as of 23:32, 10 January 2014
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Parallel_transport.png/300px-Parallel_transport.png)
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent 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, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.
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 as holonomy. The Ambrose-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 of along γ is called parallel if
Suppose we are given an element e0 ∈ EP at P = γ(0) ∈ M, rather than a section. The parallel transport of e0 along γ is the extension of e0 to a parallel section X on γ. More precisely, X is the unique section of E along γ such that
Note that in any given coordinate patch, (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–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 isomorphisms between the fibers at points along the curve:
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 the holonomy 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 the Ambrose-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 . 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 Template:Harvtxt; see Template:Harvtxt. Template:Harvtxt also adopts this approach.
Consider an assignment to each curve γ in the manifold a collection of mappings
such that
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
This defines the associated infinitesimal connection ∇ on E. One recovers the same parallel transport Γ from this infinitesimal connection.
Special case: The tangent bundle
Let M be a smooth manifold. Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called (affine) geodesics Template:Harv. A smooth curve γ: I → M is an affine geodesic if is parallel transported along , that is
Taking the derivative with respect to time, this takes the more familiar form
Parallel transport in Riemannian geometry
In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor. Thus a metric connection is any connection Γ such that, for any two vectors X, Y ∈ Tγ(s)
Taking the derivative at t=0, the associated differential operator ∇ must satisfy a product rule with respect to the metric:
Geodesics
If ∇ is a metric connection, then the affine geodesics are the usual geodesics 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 is constant on I. Indeed
It follows from an application of Gauss's Lemma that if A is the norm of then the distance, induced by the metric, between two close enough points on the curve γ, say γ(t1) and γ(t2), is given by
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 Template:Harv. Let P → M be a principal bundle over a manifold M with structure Lie 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
from the fibre over γ(s) to that over γ(t), which is an isomorphism of homogeneous spaces: i.e. for each g∈G.
Further generalizations of parallel transport are also possible. In the context of Ehresmann connections, where the connection depends on a special notion of "horizontal lifting" of tangent spaces, one can define parallel transport via horizontal lifts. Cartan connections 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.
Approximation: Schild's ladder
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Schild%27s_ladder_step_4.svg/300px-Schild%27s_ladder_step_4.svg.png)
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Parallel transport can be discretely approximated by Schild's ladder, which takes finite steps along a curve, and approximates Levi-Civita parallelogramoids by approximate parallelograms.
See also
- Basic introduction to the mathematics of curved spacetime
- Connection (mathematics)
- Development (differential geometry)
- Affine connection
- Covariant derivative
- Geodesic (general relativity)
- Lie derivative
- Schild's ladder
- Levi-Civita parallelogramoid
References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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External links
- Spherical Geometry Demo. An applet demonstrating parallel transport of tangent vectors on a sphere.