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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, vector valued functions, vector fields, or, more generally,
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. In an invariant differential operator D, the term ''
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
'' indicates that the value Df of the map depends only on f(x) and the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s of f in x. The word ''
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
'' indicates that the operator contains some
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. This means that there is a group G with a group action on the functions (or other objects in question) and this action is preserved by the operator: :D(g\cdot f)=g\cdot (Df). Usually, the action of the group has the meaning of a
change of coordinates In mathematics, an ordered basis of a vector space of finite dimension (vector space), dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a finite sequence, sequence of scalar (mathematics), ...
(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 In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
for a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
G and a Lie subgroup H. Every
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
\rho:H\rightarrow\mathrm(\mathbb) gives rise to a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
:V=G\times_\mathbb\;\text\;(gh,v)\sim(g,\rho(h)v)\;\forall\;g\in G,\;h\in H\;\text\;v\in\mathbb. Sections \varphi\in\Gamma(V) can be identified with :\Gamma(V)=\. In this form the group ''G'' acts on sections via :(\ell_g \varphi)(g')=\varphi(g^g'). Now let ''V'' and ''W'' be two
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s over ''M''. Then a differential operator :d:\Gamma(V)\rightarrow\Gamma(W) that maps sections of ''V'' to sections of ''W'' is called invariant if :d(\ell_g \varphi) = \ell_g (d\varphi). for all sections \varphi in \Gamma(V) 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 In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is t ...
s.


Invariance in terms of abstract indices

Given two
connections Connections may refer to: Television * '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series * ''Connections'' (British documentary), a documentary television series and book by science historian Jam ...
\nabla and \hat and a one form \omega, we have :\nabla_\omega_=\hat_\omega_-Q_^\omega_ for some tensor Q_^. Given an equivalence class of connections
nabla Nabla may refer to any of the following: * the nabla symbol ∇ ** the vector differential operator, also called del, denoted by the nabla * Nabla, tradename of a type of rail fastening system (of roughly triangular shape) * ''Nabla'' (moth), a ge ...
/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. Q_^=Q_^. Therefore we can compute :\nabla_\omega_=\hat_\omega_, 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
metrics Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
in the conformal class; * in projective geometry an equivalence class of connection is given by all connections that have the same
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
; * in
CR geometry In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formal ...
an equivalence class of connections is given by the Tanaka-Webster connections for each choice of pseudohermitian structure


Examples

# The usual
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
operator \nabla acting on real valued functions on
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is invariant with respect to all Euclidean transformations. # The differential acting on functions on a manifold with values in
1-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
s (its expression is
     d=\sum_j \partial_j \, dx_j
in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s is just the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...
). # More generally, the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...

     d:\Omega^n(M)\rightarrow\Omega^(M)
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 mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formall ...
in physics is invariant with respect to the Poincaré group (if we choose the proper 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 cover of the Poincaré group) # The
conformal Killing equation In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) fl ...

     X^a \mapsto \nabla_X_-\frac\nabla_c X^c g_
is a conformally invariant linear differential operator between vector fields and symmetric trace-free tensors.


Conformal invariance

Image:conformalsphere.jpg, The sphere (here shown as a red circle) as a conformal homogeneous manifold. Given a metric :g(x,y)=x_y_+x_y_+\sum_^x_y_ on \mathbb^, we can write the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
S^ as the space of generators of the nil cone :S^=\. In this way, the flat model of conformal geometry is the sphere S^=G/P with G=SO_(n+1,1) and P the stabilizer of a point in \mathbb^. A classification of all linear conformally invariant differential operators on the sphere is known (Eastwood and Rice, 1987).


See also

*
Differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s * Laplace invariant *
Invariant factorization of LPDOs The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations, which allow construction of integrable LPDEs. Laplace solved the factorization pro ...


Notes


References

* * * * {{DEFAULTSORT:Invariant Differential Operator Differential geometry Differential operators