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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.

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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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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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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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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,