|
|
| Line 1: |
Line 1: |
| In [[mathematics]] and [[theoretical physics]], an '''invariant differential operator''' is a [[Map (mathematics)|mathematical map]] from some objects to an object of similar type. These objects are typically [[Function (mathematics)|functions]] on <math>\mathbb{R}^n</math>, functions on a [[manifold]], [[vector (geometric)|vector]] valued functions, [[vector field]]s, or, more generally, [[Section (category theory)|sections]] of a [[vector bundle]].
| | Wilber Berryhill is the title his mothers and fathers gave him and he completely digs that title. My working day occupation is a travel agent. For a whilst I've been in Alaska but I will have to transfer in a yr or two. I am truly fond of handwriting but I can't make it my occupation truly.<br><br>My web blog :: [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ love psychics] |
| | |
| In an invariant differential operator <math>D</math>, the word ''differential'' indicates that the value <math>Df</math> of the image depends only on <math>f(x)</math> and the [[derivative]]s of <math>f</math> in <math>x</math>. The word ''invariant'' indicates that the operator contains some [[Symmetry in mathematics|symmetry]]. This means that there is a [[Group (mathematics)|group]] <math>G</math> that has an [[action (mathematics)|action]] on the functions (or other objects in question) and this action commutes with the action of the operator:
| |
| | |
| :<math>D(g\cdot f)=g\cdot (Df).</math>
| |
| | |
| Usually, the action of the group has the meaning of a [[change of coordinates]] (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.
| |
| | |
| ==Invariance on homogeneous spaces==
| |
| Let ''M'' = ''G''/''H'' be a [[homogeneous space]] for a [[Lie group]] G and a Lie subgroup H. Every [[Representation (mathematics)|representation]] <math>\rho:H\rightarrow\mathrm{Aut}(\mathbb{V})</math> gives rise to a [[vector bundle]]
| |
| | |
| :<math>V=G\times_{H}\mathbb{V}\;\text{where}\;(gh,v)\sim(g,\rho(h)v)\;\forall\;g\in G,\;h\in H\;\text{and}\;v\in\mathbb{V}.</math>
| |
| | |
| Sections <math>\varphi\in\Gamma(V)</math> can be identified with
| |
| | |
| :<math>\Gamma(V)=\{\varphi:G\rightarrow\mathbb{V}\;:\;\varphi(gh)=\rho(h^{-1})\varphi(g)\;\forall\;g\in G,\; h\in H\}.</math>
| |
| | |
| In this form the group ''G'' acts on sections via
| |
| | |
| :<math>(\ell_g \varphi)(g')=\varphi(g^{-1}g').</math>
| |
| | |
| Now let ''V'' and ''W'' be two [[vector bundle]]s over ''M''. Then a differential operator
| |
| | |
| :<math>d:\Gamma(V)\rightarrow\Gamma(W)</math>
| |
| | |
| that maps sections of ''V'' to sections of ''W'' is called invariant if
| |
| | |
| :<math>d(\ell_g \varphi) = \ell_g (d\varphi).</math>
| |
| | |
| for all sections <math>\varphi</math> in <math>\Gamma(V)</math> and elements ''g'' in ''G''. All linear invariant differential operators on homogeneous [[parabolic geometries]], i.e. when ''G'' is semi-simple and ''H'' is a parabolic subgroup, are given dually by homomorphisms of [[generalized Verma module]]s.
| |
| | |
| ==Invariance in terms of abstract indices==
| |
| Given two [[Connection (mathematics)|connections]] <math>\nabla</math> and <math>\hat{\nabla}</math> and a one form <math>\omega</math>, we have
| |
| :<math>\nabla_{a}\omega_{b}=\hat{\nabla}_{a}\omega_{b}-Q_{ab}{}^{c}\omega_{c}</math>
| |
| for some tensor <math>Q_{ab}{}^{c}</math>.<ref>{{cite book|author=Penrose and Rindler|title=Spinors and Space Time|publisher=Cambridge Monographs on Mathematical Physics|year=1987}}</ref> Given an equivalence class of connections <math>[\nabla]</math>, we say that an operator is invariant if the form of the operator does not change when we change from one connection in the equivalence class to another. For example, if we consider the equivalence class of all [[torsion free]] connections, then the tensor Q is symmetric in its lower indices, i.e. <math>Q_{ab}{}^{c}=Q_{(ab)}{}^{c}</math>. Therefore we can compute
| |
| :<math>\nabla_{[a}\omega_{b]}=\hat{\nabla}_{[a}\omega_{b]},</math>
| |
| where brackets denote skew symmetrization. This shows the invariance of the exterior derivative when acting on one forms.
| |
| Equivalence classes of connections arise naturally in differential geometry, for example:
| |
| | |
| * in [[conformal geometry]] an equivalence class of connections is given by the Levi Civita connections of all [[Metric (mathematics)|metrics]] in the conformal class;
| |
| * in [[projective geometry]] an equivalence class of connection is given by all connections that have the same [[geodesics]];
| |
| * in [[CR geometry]] an equivalence class of connections is given by the Tanaka-Webster connections for each choice of pseudohermitian structure
| |
| | |
| ==Examples==
| |
| # The usual [[gradient]] operator <math>\nabla</math> acting on real valued functions on [[Euclidean space]] is invariant with respect to all [[Euclidean transformation]]s.
| |
| # The [[exterior derivative|differential]] acting on functions on a manifold with values in [[one-form#Differential one-forms|1-form]]s (its expression is <br> <math>d=\sum_j \partial_j \, dx_j</math> <br>in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on [[differential form]]s is just the [[pullback (differential geometry)|pullback]]).
| |
| # More generally, the [[exterior derivative]] <br> <math>d:\Omega^n(M)\rightarrow\Omega^{n+1}(M)</math> <br>that acts on ''n''-forms of any smooth manifold M is invariant with respect to all smooth transformations. It can be shown that the exterior derivative is the only linear invariant differential operator between those bundles.
| |
| # The [[Dirac operator]] in physics is invariant with respect to the [[Poincaré group]] (if we choose the proper [[action (mathematics)|action]] of the [[Poincaré group]] on spinor valued functions. This is, however, a subtle question and if we want to make this mathematically rigorous, we should say that it is invariant with respect to a group which is a [[Double covering group|double cover]] of the Poincaré group)
| |
| # The [[conformal Killing equation]] <br> <math>X^a \mapsto \nabla_{(a}X_{b)}-\frac{1}{n}\nabla_c X^c g_{ab}</math><br> is a conformally invariant linear differential operator between vector fields and symmetric trace-free tensors.
| |
| | |
| ==Conformal invariance==
| |
| <gallery>
| |
| Image:conformalsphere.jpg| The sphere (here shown as a red circle) as a conformal homogeneous manifold.
| |
| </gallery>
| |
| Given a metric
| |
| :<math>g(x,y)=x_{1}y_{n+2}+x_{n+2}y_{1}+\sum_{i=2}^{n+1}x_{i}y_{i}</math> | |
| | |
| on <math>\mathbb{R}^{n+2}</math>, we can write the [[sphere]] <math>S^{n}</math> as the space of generators of the nill cone
| |
| | |
| :<math>S^{n}=\{[x]\in\mathbb{RP}_{n+1}\; :\; g(x,x)=0 \}.</math>
| |
| | |
| In this way, the flat model of [[conformal geometry]] is the sphere <math>S^{n}=G/P</math> with <math>G=SO_{0}(n+1,1)</math> and P the stabilizer of a point in <math>\mathbb{R}^{n+2}</math>. A classification of all linear conformally invariant differential operators on the sphere is known (Eastwood and Rice, 1987).<ref>{{cite journal|last=M.G. Eastwood and J.W. Rice|title=Conformally invariant differential operators on Minkowski space and their curved analogues|year=1987|journal=Commun. Math. Phys. 109 (1987), no. 2, 207–228}}</ref>
| |
| | |
| ==See also==
| |
| *[[Differential operator]]s
| |
| *[[Laplace invariant]]
| |
| *[[Invariant factorization of LPDOs]]
| |
| | |
| ==Notes==
| |
| <references />
| |
| | |
| ==References==
| |
| *{{cite book|last=Slovák|first=Jan|title=[ftp://www.math.muni.cz/pub/math/people/Slovak/papers/vienna.ps Invariant Operators on Conformal Manifolds]|year=1993|publisher=Research Lecture Notes, University of Vienna (Dissertation)}}
| |
| *{{cite book|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag, Berlin, Heidelberg, New York}}
| |
| *{{cite journal|last1=Eastwood|first1=M. G.|last2=Rice|first2=J. W.|title=Conformally invariant differential operators on Minkowski space and their curved analogues|year=1987|journal=Commun. Math. Phys. 109 (1987), no. 2, 207–228}}
| |
| *{{cite journal|last=Kroeske|first=Jens|title=Invariant bilinear differential pairings on parabolic geometries |year=2008|journal=Phd thesis from the University of Adelaide|arxiv=0904.3311}}
| |
| | |
| {{DEFAULTSORT:Invariant Differential Operator}}
| |
| [[Category:Differential geometry]]
| |
| [[Category:Differential operators]]
| |
Wilber Berryhill is the title his mothers and fathers gave him and he completely digs that title. My working day occupation is a travel agent. For a whilst I've been in Alaska but I will have to transfer in a yr or two. I am truly fond of handwriting but I can't make it my occupation truly.
My web blog :: love psychics