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arXiv:1110.0639v1 [math.DG] 4 Oct 2011

A RIEMANNIAN DEFINITION OF QUASIREGULAR

MAPPINGS

TONY LIIMATAINEN

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.

1. Introduction

The standard analytic definition of a quasiregular mapping is given for a

mapping φ : Ω → Rn, Ω open in Rn, by the inequality

||Dφ||n

Here Dφ is the differential of φ and

1≤ KJφ. (1.1)

||Dφ||1= sup

|X|=1

|(Dφ)X|

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. [7].

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

in Eq.1.1.

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.

1

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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 [7].

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 [5]. This definition is naturally valid also for Rie-

mannian manifolds where the distance function is given by the Riemannian

metric.

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 [12]. 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

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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.

loc(M,M′).

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

Tr?φ∗g′?n≤ K2Det?φ∗g′?

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

Tr(T) =1

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

the definition.

(2.1)

0(M) are given in local coordinates by

ntr?g−1T?

The motivation for the definition comes from the equation of a conformal

mapping,

φ∗g′= cg.

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

Tr?φ∗g′?n= Det?φ∗g′?.

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′

φ(p)).

Thus there exists the (formal) adjoint

Dφ∗: (Tφ(p)M′,g′

φ(p)) → (TpM,gp)

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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′

= ?DφT(g′

= ?(g−1)TDφTg′

φDφV ? = ?(g′

φU,g−1gV ? =

φU,gV ? = g(g−1DφTg′

φDφ)TU,V ?

φ)TU,V ? = ?DφTg′

φU,V ).

Thus the adjoint of Dφ is

Dφ∗= g−1DφTg′

φ.

The normalized Hilbert-Schmidt inner product of operators T and S be-

tween Hilbert spaces is given by

?T,S? =1

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.

ntr(T∗S).

ntr?g−1DφTg′

φDφ?= Tr?φ∗g′?.

Det(Dφ) := |

det(Dφ∗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

1

√n

?

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

definition.

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

||Dφ|| =

?

det(Dφ∗Dφ)|.

tr(Dφ∗Dφ)

Det(Dφ) = |

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS5

point p ∈ M, the mapping Dφ∗Dφ is a mapping from TpM to itself. Also a

?1

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.

Remarks. A?1

g−1T(X) = λX.

1

?-tensor can be regarded as fiber preserving linear mapping TM → TM.

The trace and determinant of linear mappings from a vector space, TpM

0

?-

tensors are independent of the local framing. In particular, the definitions

0

?-tensor, or the differ-

ential of a mapping, we mean the normalized trace and determinant unless

1

?-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

(2.3)

M′fdVg′ =

?

M

f|φDetg(Dφ)dVg. (2.4)

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

1

n||Dφ||2

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

K2Det?φ∗g′?= K2det(Dφ)2.

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-

tion.

1≤ ||Dφ||2= Tr?φ∗g′?≤ n||Dφ||2

1,

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS6

The definition of a quasiconformal mapping can also be interpreted as a

definition of quasiconformal metrics by inserting φ = Id to the definition.

We say that two Riemannian metrics g and g′are quasiconformally related

if the relation

Trg

holds for some K ≥ 1. The following proposition shows that metrics related

1-quasiconformally and 1-quasiregular mappings are conformal metrics and

conformal mappings respectively.

?g′?n≤ K2Detg

?g′?

Proposition 3.1. A 1-quasiregular mapping φ : (M,g) → (M′,g′) is con-

formal,

φ∗g′= cg,

for some positive function c.

Proof. It suffices to prove the claim locally. Any local matrix representation

of g−1φ∗g′has positive eigenvalues λias g−1φ∗g′is a product of symmetric

positive definite matrices.

Elementary calculations show that the function

Tr?φ∗g′?n− Det?φ∗g′?=

has minimum 0 for positive λi. This happens exactly when all the λi’s

coincide, λi= λ. Now, the condition of 1-quasiregularity yields

Tr?φ∗g′?n≤ Det?φ∗g′?= λ1···λn≤

Accordingly, the minimum of the function of Eq. 3.1 is achieved, and we

have

g−1φ∗g′= ATΛA = λ Id.

Here A is the diagonalizing orthogonal matrix field of the local coordinate

representation of g−1φ∗g′. The matrix Λ is the diagonal matrix, which has

the eigenvalue λ (of multiplicity n) as the diagonal entries.

?1

ordinate invariant eigenvalues for any differentiable mapping φ : (M,g) →

(M′,g′). When the mapping is K-quasiregular the ratio of the maximum

and minimum eigenvalue is in addition bounded by nnK2:

?1

n

?n

(λ1+ ··· + λn)n− λ1···λn

(3.1)

?1

n

?n

(λ1+···+λn)n= Tr?φ∗g′?n.

?

We call the

morphism. It is worth noting that the distortion tensor has positive co-

1

?-tensor g−1φ∗g′the distortion tensor or distortion endo-

K2≥ Tr?φ∗g′?n/ Det?φ∗g′?=

When the mapping φ is conformal, the only eigenvalue of the distortion

tensor is the conformal factor c of the mapping with multiplicity n.

Previously we found that quasiregular maps, as we have defined, between

Euclidean spaces are also quasiregular mappings in the traditional sense.

In Euclidean space the new and the traditional definitions are equivalent.

We consider next a local version of this equivalence for general Riemann-

ian manifolds. We wish to show that for every K-quasiregular mapping

φ : (M,g) → (M′,g′) and p ∈ M there are a constant? K and coordinate

1

nn

??

i

λi

?n

/

?

i

λi≥

1

nn

λmax

λmin

. (3.2)

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS7

neighborhoods U and V of p and φ(p) such that the coordinate representa-

tion of φ is? K-quasiregular in the traditional sense. We also wish to have a

We begin with a useful lemma, which shows that quasiregular mappings

are well behaved under composition.

good control over? K.

Lemma 3.2. Let φ1: (M1,g1) → (M2,g2) and φ2 : (M2,g2) → (M3,g3)

be K1- and K2-quasiregular mappings respectively. Then the composition

φ = φ2◦ φ1is nn/2K1K2quasiregular.

Proof. We need to evaluate the norm and the determinant of the differential

of the mapping φ = φ2◦ φ1. The differential of φ in local coordinates is

the matrix product of the Jacobian matrices of φ2evaluated at φ1and the

differential of φ1,

Dφ = (Dφ2)|φ1Dφ1.

The adjoint of Dφ is

Dφ∗= Dφ∗

Using the cyclicity of trace we have for the norm of Dφ,

1(Dφ∗

2)|φ1.

||Dφ|| = n−1/2tr((Dφ2)|φ1Dφ1Dφ∗

≤ n1/2||Dφ2||φ1||Dφ1||,

where the inequality follows from the fact that the trace is sub-multiplicative

for positive definite matrices. Similarly, we see that the determinant is

multiplicative,

1(Dφ∗

2)|φ1)1/2

Det(Dφ) = Det(Dφ2)|φ1Det(Dφ1).

The claim follows now immediately from these equalities and definitions:

||Dφ||n≤ nn/2||Dφ2||n

(3.3)

φ1||Dφ1||n≤ nn/2K1K2Det(Dφ).

?

With the composition rule above, we can easily prove that quasiregular

mappings are locally quasiregular mappings in the traditional sense. We

begin with a definition.

Definition 3. Let φ be a mapping between Riemannian manifolds (M,g)

and (M′,g′). The distortion function K2of φ is

Tr(φ∗g′)n

Det(φ∗g′)(p).

K2(p) =

Note that the mapping φ is quasiregular exactly when the distortion func-

tion is bounded in M. The distortion function is invariant under conformal

scaling of either or both metrics g and g′. Thus we observe that the notion

of quasiregularity is naturally conformally invariant.

Theorem 3.3. Let φ : (M,g) → (M′,g′) be a K-quasiregular mapping,

and p ∈ M. Then there are coordinate neighborhoods U and V for p and

φ(p) such that the coordinate representation of the restriction of φ : U → V

satisfies the traditional equation of quasiregularity with a quasiconformal

factor? K depending on K and the coordinate representations of g in U and

g′in V.

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS8

Proof. Let p ∈ M.

V around p and φ(p) such that φ(U) ⊂ V. We can also assume that the

coordinate representations of g and g′are continuous on U and V. We

estimate the distortion function of the mapping

Choose (bounded) coordinate neighborhoods U and

˜φ = Id ◦ φ ◦ Id : (U,e) → (U,g) → (V,g′) → (V,e).

Here we consider U and V to be open sets in Rn. Interpreting the coordinates

of (U,e) and (V,e) as Cartesian coordinates, where e is the identity matrix,

the distortion functions of the identity maps in the composition above are

K2

1= Tre(Id∗g)n/ Dete(Id∗g) =

1

nntr(g)n/ det(g)

1

nntr?g′−1?n/ det?g′−1?.

K2

2= Trg′ (Id∗e)n/ Detg′ (Id∗e) =

Now, using the observation that a K-quasiregular map between Euclidean

domains is nn/2K-quasiregular in the traditional sense and the previous

lemma, we see that the mapping φ|U =˜φ : U → V is quasiregular in the

traditional sense with a quasiconformality constant

? K2= n3nK2max

p∈U

K2

1(p) max

p∈φ(U)K2

2(p).

?

We end this section with an example.

Example. Let X be a vector field on (M,g). We study the quasiconfor-

mality of the flow φtgenerated by X by calculating the time derivative of

the distortion function K(t).

Denote g(t) = φ∗

tg. We take traces and determinants with respect to

g unless otherwise indicated.Elementary calculations together with the

identity

d

dtDet(g(t)) = Det(g(t)) Trt(˙ g(t))

yield

Tr(g(t))n

Det(g(t))= nK(t)2

d

dtK(t)2=d

dt

?Tr(˙ g(t))

Tr(g(t))− Trt(˙ g(t))

?

.

We can write the equation as

˙K(t) = K(t)

n/2

Tr(g(t))Tr?˙ g(t) − Trt(˙ g(t))g(t)?. (3.4)

Let h be any Riemannian metric on M, T ∈ T2

norm in the bundle T ∈ T2

||T||2

an inequality

0(M) and || · ||hthe usual

?2

0(M) induced by the Riemannian metric h,

h= tr?h−1TTh−1T?. The invariant trace of a

Trg(T) ≤ Trg(h)||T||h.

To see this, use Cauchy-Schwartz inequality for the Hilbert-Schmidt inner

product of the representation matrices of g−1and T in a h-orthonormal

0

?-tensor T satisfies

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS9

frame,

n Trg(T) = tr?g−1T?≤ ||g−1||HS||T||HS= tr?g−2?1/2||T||HS

≤ tr?g−1?||T||HS= n Trg(h)||T||h.

Applying the inequality with h = g(t) to Eq. 3.4 gives an estimate

˙K(t) ≤ K(t)n

The Riemannian inner product, norm and the invariant trace of any tensor

T behave naturally under pullbacks,

2||˙ g(t) − Trt(˙ g(t))g(t)||g(t).

||T||g|φ= ||φ∗T||φ∗g, Trg(T)|φ= Trφ∗g(φ∗T).

Also, the time derivative of g(t) is the pullback by φtof the Lie derivative

of g in the direction of X (cf. [2, p. 13]),

˙ g(t) = φ∗

tLXg = φ∗

t˙ g(0).

With these identities we can write

||˙ g(t) − Trt(˙ g(t))g(t)||g(t)= ||φ∗

tLXg − Trφ∗

= ||LXg − Tr(LXg)g||g|φt= ||SX||g|φt.

tg(φ∗

tLXg)φ∗

tg||φ∗

tg

The notation SX refers to the Ahlfors operator, whose kernel defines con-

formal Killing vector fields [11]. Together with Gr¨ onwall’s inequality we see

that the quasiconformality constant under the flow of the vector field X is

controlled in terms of the supremum g-norm of SX:

K(t) ≤ exp(tn||SX||∞/2).

The flow of X is a family of K(t)-quasiconformal mappings with K(t) given

above. We have recovered an analogous result to the one given in [11].

4. Invariant conformal structure of a quasiconformal group

In this section, we demonstrate how to use the new definition of qua-

siconformal mappings on Riemannian manifolds we have given. We study

a quasiconformal group G of homeomorphisms on a Riemannian manifold

and find an invariant conformal structure for the group. The result is a

generalization of a result by Tukia [12]:

Theorem. Let G be a group of quasiconformal mappings φ : U → U,

U ⊂ Rn∪ {∞} open. Then there is a G-invariant conformal structure µ on

U.

A K-quasiconformal group of mappings consists of quasiconformal map-

pings with the same quasiconformal factor K. An invariant conformal struc-

ture for a quasiconformal group on Rnis a positive definite symmetric matrix

field µ(x) ∈ Rn×n, x ∈ Rn, such that the mappings of the group G satisfies

the Beltrami equation

Dφ(x)Tµ(φ(x))Dφ(x) = | det(Dφ)|2/nµ(x).

Here φ ∈ G and the equation is assumed to hold a.e. x ∈ Rn. Moreover, the

determinant of the conformal structure µ is assumed to equal 1.

(4.1)

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 10

On a Riemannian manifold (M,g), we mean by a conformal structure a

Riemannian metric whose g-invariant determinant equals 1. Riemannian

metrics we consider are not assumed to be smooth nor continuous, and they

can be defined only a.e. A conformal structure h is an invariant conformal

structure for a quasiconformal group G if every mapping of the group is a

conformal mapping with respect to h,

φ∗h(p) = c(p)h(p) a.e. p ∈ M, φ ∈ G.(4.2)

The conformal factor c = cφcan be solved to give

c = Deth(φ∗h)1/n= Detg(φ∗g)1/n≡ Detg(Dφ)2/n.

Here the second equality holds by the assumption that the invariant deter-

minant of h equals 1. We will prove the following theorem.

(4.3)

Theorem 4.1. Let (M,g) be a manifold with smooth Riemannian met-

ric g, and G a countable quasiconformal group of homeomorphisms on the

manifold. Then there is a Riemannian metric h, which is measureable with

respect to measure given by g, such that the equation

φ∗h = ch

(4.4)

holds a.e. p ∈ M and for all mappings φ of the group G. The g-invariant

determinant of h equals 1. The Riemannian metric h is called an invariant

conformal structure for the group G.

A generalization of the theorem to uncountable groups should be straight-

forward by following the arguments in Tukia’s work [12]. The theorem of

Tukia has also been generalized to abelian quasiregular semigroups, which

includes uniformly quasiregular mappings as a special case making the ger-

alization especially interesting [8, 7]. The additional arguments needed to

pass from groups to abelian semigroups are local ones. Hence, we expect

Theorem 4.1 to generalize to abelian quasiregular semigroups also in the

general Riemannian setting.

Conformal structures are sections of the bundle S of positive definite

symmetric 2-covariant tensors, whose invariant determinant equals one. The

bundle S admits a special geometry, which we have to construct before the

proof of Theorem 4.1.

4.1. The space S. Let (M,g) be a Riemannian manifold and T2

bundle of its 2-covariant tensors, the tensors with two lower indices. We

consider the subset S of the bundle T2

definite tensors A whose invariant determinant,

Det(A) = det?g−1A?,

equals one. We give S a fiber bundle structure over M, which will have

the following properties. The fibers of S are naturally diffeomorphic to

the manifold P of positive definite symmetric determinant one matrices. A

transition function of the bundle S can be taken to be a mapping from an

open subset of M to the set of orthogonal matrices O(n). Thus the structure

group of S is O(n). The construction of the fiber bundle here follows an

accessible introduction to fiber bundles of [4].

0M the

0M that consists of symmetric positive

(4.5)

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 11

Let us start by constructing local trivializations for S. For this, choose

a (smooth) orthonormal coframe {ei} on an open subset U of M. We can

write an element A ∈ T2

A = Aijei⊗ ej.

Here the components Aijdefine a positive definite symmetric matrix. In an

orthonormal coframe, it also holds that det(A) = det?g−1A?= Det(A) =

mapping

ψ : π−1U → U × P,

where π is the restriction of the bundle projection T2

If (˜ψ,V) is another trivialization corresponding to an orthonormal coframe

{˜ ei}, we have a representation for A ∈ π−1(U ∩ V) as

A =˜Aij˜ ei⊗ ˜ ej.

Since any two orthonormal basis are related by an orthogonal transforma-

tion, it follows that

0U ∩ S as

(4.6)

1. Thus A can be identified with a matrix of P. A local trivialization is a

A ?→ (π(A),Aij), (4.7)

0M → M to S → M.

(4.8)

A =˜Aij˜ ei⊗ ˜ ej=˜Aij(hei) ⊗ (hej) =˜Aijhi

where h = (hi

j) ∈ O(n) is an orthogonal matrix. Therefore, the matrices

(Aij) and (˜ Aij) representing the same A ∈ S are related by

A = hT˜Ah.

kek⊗ hj

lel, (4.9)

(4.10)

A transition function cUV at p ∈ U ∩ V is a diffeomorphism of P. By the

calculation above, we have

cUV(p)[A] = h(p)TAh(p),

where h : U ∩ V → O(n). Thus S is an O(n) fiber bundle over (M,g) with

fibers diffeomorphic to P. We call P the typical fiber of S and a fiber over

p ∈ M is denoted by Sp.

The set P of positive definite symmetric matrices with determinant 1 is

a smooth manifold. It can be identified as

SL(n)/SO(n). (4.11)

We equip P with a Riemannian metric gPdefined by

gP

A(U,V ) := tr?A−1UA−1V?,

The form gPis indeed a Riemannian metric. It is bilinear and symmetric.

The positive definiteness can be seen by writing

A(U,U) = tr?A−1UA−1U?= ||A−1/2UA−1/2||2

Here ||·||HSis the usual Hilbert-Schmidt norm of matrices. The equality fol-

lows from the fact that a positive definite symmetric matrix has a symmetric

square root and the fact that the tangent vectors of symmetric matrices are

symmetric.

The Riemannian metric gPis invariant under the action

A ∈ P,U,V ∈ TAP. (4.12)

gP

HS. (4.13)

X[A] = | det(X)|−2/nXTAX,X ∈ GL(n), A ∈ P(4.14)

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS12

of the general linear group GL(n). Geometrically (P,gP) is a complete,

simply connected, globally symmetric Riemannian manifold of negative cur-

vature [12, 6].

The geometry of P extends naturally to the fibers of S. Let us first con-

struct a fiber metric for S. A fiber metric of a fiber bundle is a Riemannian

metric for each fiber π−1(p). It is an inner product for tangent vectors

U,V ∈ TAπ−1(p), where A ∈ π−1(p).

For the construction of the fiber metric, let p ∈ M, A ∈ π−1(p) and

U,V ∈ TAπ−1(p). By definition, tangent vectors U and V of TAπ−1(p) are

given by paths A1(t) and A2(t) through A in π−1(p). In a local trivialization,

we have

A1(t) = (π(π−1(p)),A1ij(t)) = (p,Aij+ t˙A1ij(0) + O(t2))

A2(t) = (π(π−1(p)),A2ij(t)) = (p,Aij+ t˙A2ij(0) + O(t2)).

We define an inner product for U and V using the local trivialization as

?

It is the inner product gPof the representation matrices.

The inner product of (4.15) is well defined. If˜A,˜ A1(t) and˜A2(t) corre-

spond to another trivialization, we have

?U,V ?A:= tr

A−1˙A1(0)A−1˙A2(0)

?

.

(4.15)

A−1= (hT˜Ah)−1= h−1˜A−1h−T

˙A1(0) = hT ˙˜A1(0)h

˙A2(0) = hT ˙˜A2(0)h

yielding

?U,V ?A= tr

??

?

h−1˜A−1h−T??

˜A−1 ˙˜A1(0)˜ A−1 ˙˜A2(0)

hT ˙˜A1(0)h

?

??

h−1˜A−1h−T??

hT ˙˜A2(0)h

??

= tr.

Thus the definition of the inner product calculated in different local trivial-

izations of π−1U and π−1V agree in the overlap of U and V. Fibers of S are

isometric to the typical fiber (P,gP).

We denote the fiber metric by gV, where V indicates that it is an inner

product for the vertical vectors of S. The fiber distance dVis induced by

the fiber metric gV. In a local trivialization, it holds, with a slight abuse of

notation, that

dV(A,B) = dP(A,B). (4.16)

Here A and B on the right hand side denotes the corresponding representa-

tions of A and B in the local trivialization. On the left hand side A and B

are elements of a fiber of S.

We define the determinant normalized pullback φ∗

ferentiable mapping φ : M → M. It is the pullback of 2-covariant tensors

whose determinant is normalized by the formula

Nof an invertible dif-

φ∗

NA :=

φ∗A

Det(φ∗g)1/n. (4.17)

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 13

The invariant determinant of a tensor A ∈ Sφis preserved

det?g−1DφTADφ?

For each p ∈ M, the normalized pullback is a mapping from the fiber of S

over φ(p) to the fiber of S over p.

Finally, we check that the normalized pullback in an isometry between

the fibers Sφ(p)and Spof S equipped with the fiber metric gVfor any p ∈ M.

To simplify the notation, we denote for a while F = φ∗

this notation, the isometry condition for φ∗

Det(φ∗

NA) =

det(g−1DφTg|φDφ)= det?g−1|φA?= 1.

N: Sφ(p)→ Sp. With

Nreads

gV

F(A)(F∗U,F∗V ) = gV

A(U,V ), (4.18)

where F∗is the pushforward of F and U,V ∈ TASφ(p), A ∈ Sφ(p). To verify

this, choose paths U(t),V (t) through A to which U and V are tangent at

A. We have

DφTU(t)Dφ

det(g−1φ∗g)1/n=

F∗U =d

dt|t=0φ∗

NU(t) =d

dt|t=0

DφTUDφ

det(g−1φ∗g)1/n.

and similarly for V . This gives

gV

F(A)(F∗U,F∗V ) = tr

?

DφTADφ

det(g−1φ∗g)1/n

?−1

DφTUDφ

det(g−1φ∗g)1/n

×

= tr?A−1UA−1V?= gV

?

DφTADφ

det(g−1φ∗g)1/n

?−1

DφTV Dφ

det(g−1φ∗g)1/n

A(U,V )

(4.19)

showing that the isometry condition holds.

4.2. Proof of the theorem 4.1. We now prove Theorem 4.1. We study a

set valued section X of S constructed from the normalized pullbacks of the

mappings of the group G applied to the given Riemannian metric g:

X(p) = {ψ∗

Ng(p) : ψ ∈ G}.(4.20)

In case the initial metric g happens to be an invariant conformal structure

X is just the initial metric g. Otherwise the quasiconformality condition will

restrict the values of X such that each set X(p) ⊂ Spbelongs to a unique

smallest ball in the topology given by the fiber metric of S. The section of S

consisting of the centers of those balls is shown to be an invariant conformal

structure. The proof is a generalization of the mentioned Tukia’s result [12,

Theorem F].

Proof of Theorem 4.1. Let G be a countable K-quasiconformal group of

homeomorphisms on a Riemannian manifold (M,g). By Theorem 3.3 the

mappings of the group are locally quasiconformal in the traditional sense

and thus differentiable a.e. [13]. The countability of G implies that we can

find a G-invariant subset M′⊂ M with the following properties. The set

M′is measurable and is of full measure, and every φ ∈ G is differentiable

Page 14

A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 14

with a non-vanishing Jacobian determinant. By the properties of M′, we

can define a set valued section X : M′→ S by

X(p) = {ψ∗

We show next that the quasiconformality of the mappings of the group G

will give a uniform bound for the diameter of each set X(p) ⊂ Sp. To show

this, let p ∈ M′and φ ∈ G be arbitrary, and choose a local trivialization

for a neighborhood of p as in (4.7). We calculate the fiber distance between

φ∗

Ng and g at p. In a local trivialization, we have by (4.16) that

dV

Ng(p) : ψ ∈ G}.

p(g(p),φ∗

Ng(p)) = dP(g(p),φ∗

Ng(p)), (4.21)

where on the right hand side we have the distance in the typical fiber P of

the representation matrices.

In the local trivialization of S, the representation matrix of g is just the

identity matrix I. For this reason, the representation matrix of φ∗g is the

same as the representation matrix of g−1φ∗g. Hence, for the distance (4.21)

it holds that

dV

p(g(p),φ∗

Ng(p)) = dP(I,g−1(p)φ∗

Ng(p)). (4.22)

It is shown in [9, p. 27] that the distance of an element A ∈ P from the

identity matrix I is given by

dP(I,A) = ((log µ1)2+ ··· + (logµn)2)1/2,

where µiare the eigenvalues of A. Let us denote by µithe eigenvalues of

g−1(p)φ∗

Ng(p). We observed in (3.2) that the ratio of the largest and smallest

eigenvalue of g−1φ∗g is bounded by nnK2. Of course, the same bound holds

for all the ratios λi/λjof the eigenvalues λiof g−1φ∗g, i,j = 1,...,n. Thus

it holds that

µn

i=

λ1···λn

Finally, we have

(4.23)

λn

i

≤ (nnK2)n. (4.24)

dV

p(g(p),φ∗

Ng(p) = ((logµ1)2+ ··· + (logµn)2)1/2≤ n1/2lognnK2. (4.25)

Since φ ∈ G and p ∈ M were arbitrary, the diameter of each set X(p) ⊂ Sp

is uniformly bounded by 2n1/2lognnK2.

Next we will show that X is invariant under the normalized pullbacks of

the group G. Normalized pullbacks satisfy a product rule

ψ∗

N◦ φ∗

N= (φ ◦ ψ)∗

N,

which follows from the chain rule of the invariant determinant

Detg(φ∗g)|ψDetg(ψ∗g) = Detg((φ ◦ ψ)∗g)

shown in (3.3). By this product rule and the group property of G, we see

that

(φ∗

NX)(p) = {φ∗

for all φ ∈ G showing the invariance of X. In this sense, X can be viewed

as a set valued solution to the problem of finding an invariant conformal

structure for G.

N(ψ∗

Ng)(p) : ψ ∈ G} = {(ψ ◦ φ)∗

Ng(p) : ψ ∈ G} = X(p),

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 15

It is shown in [12] that each bounded set of P belongs to a unique smallest

ball. Since the fibers of S are isometric to P, it follows that each X(p)

belongs to a unique smallest ball. We denote by h the section formed from

the centers of the smallest balls.

We have seen that X is invariant under normalized pullbacks of the group

and recall from (4.18) that the normalized pullback is an isometry between

the fibers of S.

centers, is mapped to each other by the normalized pullback. Accordingly,

Thus the unique smallest balls, and in particular their

φ∗

Nh(p) = h(p), (4.26)

for every p ∈ M and φ ∈ G. This means that h is an invariant conformal

structure for the group G.

Measurability of h follows by exactly the same argument as in the proof

of the theorem by Tukia [12, Theorem F]. Let G = {φ0,φ1,...} and ap-

proximate X by Xj(p) = {φ∗

the mapping H, which maps Xjto the section constructed from the centers

of unique smallest balls. Since H is continuous in the Hausdorff metric, it

follows that each hjis measurable and that hj→ h as j → ∞. Hence h is

measurable.

iNg : i ≤ j}. Consider hj to be the image of

?

Acknowledgements. The author is partly supported by Finnish National

Graduate School in Mathematics and its Applications. The author would

like to thank Kirsi Peltonen for helpful suggestions.

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A RIEMANNIAN DEFINITION OF QUASIREGULAR MAPPINGS 16

Department of Mathematics and Systems Analysis, Aalto University, P.O.

Box 11100 FI-00076 Aalto Finland, +358 9 470 23044

E-mail address: tony.liimatainen@tkk.fi