In
mathematics, Liouville's theorem, proved by
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
in 1850, is a
rigidity theorem about
conformal mapping
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
s in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
. It states that any
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
conformal mapping on a domain of R
''n'', where ''n'' > 2, can be expressed as a composition of
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
similarities,
orthogonal transformations and
inversions: they are
Möbius transformations (in ''n'' dimensions).
[ Philip Hartman (1947]
Systems of Total Differential Equations and Liouville's theorem on Conformal Mapping
American Journal of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press.
History
The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
69(2);329–332. This theorem severely limits the variety of possible conformal mappings in R
3 and higher-dimensional spaces. By contrast, conformal mappings in R
2 can be much more complicated – for example, all
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...
planar domains are
conformally equivalent, by the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic ...
.
Generalizations of the theorem hold for transformations that are only
weakly differentiable . The focus of such a study is the non-linear
Cauchy–Riemann system that is a necessary and sufficient condition for a smooth mapping ''ƒ'' → Ω → R
''n'' to be conformal:
:
where ''Df'' is the
Jacobian derivative, ''T'' is the
matrix transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
, and ''I'' is the identity matrix. A weak solution of this system is defined to be an element ''ƒ'' of the
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
''W''(''Ω'',R
''n'') with non-negative Jacobian determinant
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form
:
where ''a'',''b'' are vectors in R
''n'', α is a scalar, ''A'' is a rotation matrix, ε = 0 or 2, and the matrix in parentheses is ''I'' or a
Householder matrix
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformat ...
(so, orthogonal). Equivalently stated, any
quasiconformal map
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.
Intuitively, let ''f'' : ''D' ...
of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space ''W''
1,''n'', since ''ƒ'' ∈ ''W''(''Ω'',R
''n'') then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space. The result is not optimal however: in even dimensions ''n'' = 2''k'', the theorem also holds for solutions that are only assumed to be in the space ''W'', and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in ''W''
1,''p'' for any ''p'' < ''k'' which are not Möbius transformations. In odd dimensions, it is known that ''W''
1,''n'' is not optimal, but a sharp result is not known.
Similar rigidity results (in the smooth case) hold on any
conformal manifold
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
. The group of conformal isometries of an ''n''-dimensional conformal
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
always has dimension that cannot exceed that of the full conformal group SO(''n''+1,1). Equality of the two dimensions holds exactly when the conformal manifold is isometric with the
''n''-sphere or
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
. Local versions of the result also hold: The
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of
conformal Killing fields in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.
Notes
References
* .
*
Harley Flanders
Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory, linear algebra, electrical networks, scientific computing.
...
(1966) "Liouville's theorem on conformal mapping",
Journal of Mathematics and Mechanics 15: 157–61,
*
* .
* .
* {{springer, id=L/l059680, title=Liouville theorems, first=E.D., last=Solomentsev, year=2001
Conformal mappings
Theorems in geometry