arXiv:1110.0639v1 [math.DG] 4 Oct 2011
A RIEMANNIAN DEFINITION OF QUASIREGULAR
Abstract. In this paper, we give a new natural coordinate invariant
definition of quasiconformal and quasiregular mappings on Riemannian
manifolds. The new definition is given in terms of coordinate invariant
trace and determinant. The definition can also be seen arising naturally
from the inner product space structure the tangent spaces of Riemannian
manifolds have. In case of Euclidean spaces, the definition is equivalent
to the standard analytic definition of quasiregular mappings. We demon-
strate how to use the new definition by finding an invariant conformal
structure for quasiconformal groups on Riemannian manifolds.
The standard analytic definition of a quasiregular mapping is given for a
mapping φ : Ω → Rn, Ω open in Rn, by the inequality
Here Dφ is the differential of φ and
is its operator norm. The norms of the vectors are given by Euclidean inner
product. The quasiconformality constant K is assumed to be greater than
or equal to 1 and Jφis the Jacobian determinant of φ. The mapping φ is
naturally considered to belong to the Sobolev space W1,n
is assumed to hold a.e. .
The definition does not directly generalize to mappings between mani-
folds. The problem is that the Jacobian determinant of a mapping depends
not only on the point, but also on the chosen coordinates. It is not coordi-
nate invariant. On manifolds, which are only locally pieces of Rn, we do not
have canonical global coordinates to make sense of the Jacobian determinant
Let us reason why the Jacobian determinant is coordinate dependent.
The Jacobian matrix of a mapping depends on the coordinates of both the
domain and target spaces. A coordinate transformation on either space
accounts for multiplication of the Jacobian matrix by the coordinate trans-
formation matrix. Thus, the Jacobian determinant in the new coordinates is
dependent also on the determinant of the coordinate transformation matrix
illustrating the coordinate dependence.
loc(Ω,Rn) and Eq. 1.1
Date: October 4, 2011.
A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS2
The source of the coordinate dependence is in that Dφ at a point p ∈ M
is a linear mapping between the tangent spaces TpM and Tφ(p)M. On the
other hand, determinant is consistently defined only for linear mappings
from a vector space to itself. As the tangent spaces of Rncan be identified,
there is no inconsistency within Eq. (1.1) in Rn.
Defining quasiregular mappings for manifolds is not new. The common
well studied definition is to construct suitable coordinate charts in which
the distortion from the choice of coordinates to Eq. 1.1 remains bounded .
A collection of such coordinate charts is called a quasiconformal structure.
This definition has been successfully applied in the solutions of several prob-
lems [10, 1]. The benefit of this definition is that it assumes minimal struc-
ture for the manifold. Except in dimension 4, quasiconformal structures can
be found even for topological manifolds [7, 3].
The other common definition is the metric definition of quasiregular map-
pings on metric spaces . This definition is naturally valid also for Rie-
mannian manifolds where the distance function is given by the Riemannian
We give a new definition for quasiregular mappings between Riemannian
manifolds. It has two advantages over the previous ones. The definition is
independent of the choice of coordinates and is written in the language of
tensor analysis. Unlike the metric definition of quasiregular mappings, the
definition we give is analytic. That is, it is given by the derivatives of the
mapping. Moreover, we have the luxury to make any choice of local coordi-
nates for calculations, which is absent in the first definition of quasiregular
mappings on manifolds described above. The freedom to choose coordinates
might be of interest also for the study of quasiregular mappings in Rn.
The new definition of quasiregular mappings and the basic properties of
mappings satisfying the new definition are the main subjects of this paper.
In addition, we give an application for the definition. As an application we
generalize a result by Tukia concerning invariant conformal structures for
quasiconformal groups . We show that every (countable) quasiconformal
group on a Riemannian manifold admits an invariant conformal structure.
Previously this result has been proved only for Rnand n-spheres. By us-
ing the definition we give for quasiconformality, Tukia’s result generalizes
naturally for arbitrary Riemannian manifolds.
To prove the existence of invariant conformal structure for quasiconformal
group, one has to construct a fiber bundle over the manifold whose sections
are conformal structures. The construction of this bundle is also done in
this paper and we show that the bundle admits an elegant geometry. The
geometry of the bundle is inherited form the natural geometry of positive
definite symmetric determinant one matrices. The bundle might a useful
concept in its own in other contexts as well.
2. A Riemannian definition of quasiregular mappings
The coordinate invariant definition of quasiregular and quasiconformal
mappings is given via the Riemannian metric. The regularity assumptions
of the Riemannian metrics and mappings play only a small role in the first
A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS3
two sections. The definitions and results given in the first two sections are
valid for smooth C∞mappings and Riemannian metrics. It can be seen from
the text that, with slight modifications, we can also consider for example
measurable Riemannian metrics and mappings belonging to W1,n
In the last section of this paper, we consider measurable Riemannian metrics
and a case where the mappings are only homeomorphisms.
Definition 1. (The invariant definition) Let M and M′be Riemannian
manifolds with metrics g and g′. A mapping φ : M → M′is said to be
K-quasiregular if there is a number K ≥ 1, for which the inequality
holds. Here φ∗g′is the pullback of g′by φ. If φ is in addition a homeomor-
phisms, we call it a K-quasiconformal mapping.
The invariant normalized trace Tr and the invariant determinant Det for
a general 2-covariant tensor T ∈ T2
Det(T) = det?g−1T?.
Here trace and determinant on the right hand sides are the usual ones for
the matrix product g−1T of the representation matrices of tensors g−1and
T with respect to any local frame and coframe. Occasionally we also use
notations Trg(·) and Detg(·) to emphasize the Riemannian metric used in
0(M) are given in local coordinates by
The motivation for the definition comes from the equation of a conformal
The conformal factor c can be solved from the equation in several ways. By
taking the invariant normalized trace and invariant determinant, one has
c = Tr?φ∗g′?
Equating the formulas for c yields
Relaxation of this equality to an inequality with a factor K2≥ 1 gives the
definition of quasiregular mappings. A conformal mapping is 1-quasiregular,
and the converse is also true as is shown in Proposition 3.1. The appearance
of the square of K will become apparent later.
There is a very natural way to interpret the new definition of quasiregular
mappings. The tangent spaces of Riemannian manifolds M and M′are
equipped with inner products g and g′. Thus the differential of φ at p ∈ M
is a linear mapping between inner product spaces,
and cn= Det?φ∗g′?.
Dφ : (TpM,gp) → (Tφ(p)M′,g′
Thus there exists the (formal) adjoint
φ(p)) → (TpM,gp)
A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS4
of Dφ satisfying
g(Dφ∗U,V ) = g′
φ(U,DφV ), (2.2)
for all U ∈ Tφ(p)M′and V ∈ TpM at each point p ∈ M. The differential
of φ and its adjoint are well defined not only pointwise, but also as bundle
maps between (TM,g) and (TM′,g′). We calculate next the explicit form
of the adjoint.
Choose p ∈ M and local frames on some neighborhoods of p ∈ M and
φ(p) ∈ M′, and denote by ?·,·? the standard Euclidean inner product of
vectors. From the definition of the adjoint we calculate
g(Dφ∗U,V ) = g′
φ(U,DφV ) = ?U,g′
φDφV ? = ?(g′
φU,g−1gV ? =
φU,gV ? = g(g−1DφTg′
φ)TU,V ? = ?DφTg′
Thus the adjoint of Dφ is
The normalized Hilbert-Schmidt inner product of operators T and S be-
tween Hilbert spaces is given by
The induced norm of the normalized Hilbert-Schmidt inner product applied
to Dφ now yields
||Dφ||2= ?Dφ,Dφ? =1
This is the natural interpretation of the invariant trace in Definition 1. It
is the normalized Hilbert-Schmidt norm of Dφ : (TM,g) → (TM′,g′). We
also define yet another determinant as the square root of Det(φ∗g′),
Using what we have just observed, we give an equivalent definition of
quasiregular mappings. In the sense that it is given in terms of natural
mappings between tangent spaces, it is more geometrical than the first one.
Det(Dφ) := |
Definition 2. (Equivalent definition) A mapping (homeomorphism) φ :
(M,g) → (M′,g′) is said to be K-quasiregular (K-quasiconformal) if
||Dφ||n≤ K Det(Dφ).
Here ||Dφ|| and Det(Dφ) are the coordinate invariant normalized Hilbert-
Schmidt norm and the determinant defined as
The definition is just the (positive) square root of Definition 1 of quasireg-
ular mappings. It also explains the appearance of K squared in the first
We have now defined two traces and determinants, which we have claimed
to be coordinate invariant. To see the invariance, note first that, at each
Det(Dφ) = |
A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS5
point p ∈ M, the mapping Dφ∗Dφ is a mapping from TpM to itself. Also a
in this case, to itself are independent of the choice of the basis. Thus, the
defined traces and determinants for Dφ, g−1φ∗g′and more generally for?2
are independent of the choice of a local coordinate frame.
From now on, by a trace and a determinant of a?2
otherwise stated. The normalized Hilbert-Schmidt norm, denoted simply as
|| · ||, will be our default norm for Dφ.
g−1T(X) = λX.
?-tensor can be regarded as fiber preserving linear mapping TM → TM.
The trace and determinant of linear mappings from a vector space, TpM
tensors are independent of the local framing. In particular, the definitions
?-tensor, or the differ-
ential of a mapping, we mean the normalized trace and determinant unless
?-tensor g−1T, T ∈ T2
0(M), thought as a linear bundle map
TM → TM, has also a well defined eigenvalue equation
The coordinate independent eigenvalues, which naturally depend on the
point of the manifold, are solutions to the coordinate invariant equation
Det(T − λg) = det?g−1T − λI?= 0.
The invariant determinant of a diffeomorphism φ : (M,g) → (M′,g′) also
arises from the integration by substitution formula in the sense that
3. Basic properties of quasiregular mappings
The new Riemannian definition of quasiregular mappings is a natural
definition of quasiregular mappings in the Riemannian setting. In addition,
it generalizes the traditional definition (1.1) of quasiregular mappings in Rn.
To see this, let the manifolds in the definition be Euclidean domains in Rn
with Cartesian coordinates. In these domains the metrics g and g′are the
identity matrices I. Now, for the normalized invariant trace in the defining
equation (2.1) holds
where we have denoted the operator and normalized Hilbert-Schmidt norms
by || · ||1and || · || respectively. Moreover, the right hand side of (2.1), for
g = g′= I, now reads
Together these show that a K-quasiregular mapping in the traditional sense,
as defined by Eq. 1.1, is nn/2K-quasiregular mapping in the new sense. On
the contrary, a K-quasiregular mapping between Euclidean domains in Rnin
the new sense is also a nn/2K-quasiregular mapping in the traditional sense.
The quasiregularity constants nn/2K in these cases are not the best ones in
general. For example, 1-quasiregular mappings in Rnare 1-quasiregular in
both new and old definitions, which can be deduced from the next proposi-
1≤ ||Dφ||2= Tr?φ∗g′?≤ n||Dφ||2