#### Abstract

We show that 1-quasiconformal mappings on Goursat groups are CR or anti-CR mappings. This can reduce the determination of 1-quasiconformal mappings to the determination of CR automorphisms of CR manifolds, which is a fundamental problem in the theory of several complex variables.

#### 1. Introduction

The well-known Liouville theorem tells us that 1-quasiconformal mappings on are elements of group . Quasiconformal mappings on the Heisenberg group were introduced by Mostow [1] in his study of rigidity theorem. Since then, quasiconformal mappings on nilpotent Lie group and on more general Carnot-Carathéodory spaces had drawn more and more concern. Korányi and Reimann [2] and Korányi and Reimann [3] showed that orientation preserving 1-quasiconformal mappings between domains of Heisenberg group are CR or anti-CR mappings and are obtained by the restriction of group actions (see also [4]). Cowling et al. [5] extended Liouville theorem to conformal mappings on , where is a semisimple Lie group and is a minimal parabolic subgroup of . Ottazzi and Warhurst [6, 7] obtained a Liouville type theorem for all Carnot groups which sated that 1-quasiconformal maps form finite dimensional Lie groups. Xie [8] investigated 1-quasiconformal mappings on Carnot groups with reducible first layer equipped with left-invariant sub-Riemannian Carnot metrics and showed that each 1-quasiconformal map defined on a domain of such groups is the restriction of the composition of a standard dilation, a left translation, and an isometric graded isomorphism. In [9], we showed the CR property of 1-quasiconformal mappings on the Engel group and obtained the identity component of the group of 1-quasiconformal mappings. For other deformations of CR mappings between the nilpotent Lie group of step two and their Beltrami equations, one can refer to [10]. In this paper, we will investigate the 1-quasiconformal mappings on Goursat groups, which are stratified nilpotent Lie groups including Engel groups and a large class of high step Carnot groups.

A* Goursat group* is a nilpotent group endowed with a Goursat distribution . A distribution of corank on a manifold is* Goursat* if the subsheaves of the tangent bundle defined inductively by ( denotes the sheaf of vector fields generated by and the Lie brackets ) correspond to distributions; that is, they have constant rank, and this rank is , [11]. By a distribution of rank we understand any subbundle of linear dimension in the tangent bundle ; its corank is .

Our model of Goursat groups is in [12], which takes as the underlying space with the group multiplication It is easy to see that its Lie algebra of left-invariant vector fields is given by Then span the tangent space everywhere and the only nontrivial commutation relations among these vector fields are We use to denote the Lie algebra and it splits into the direct sum where the vector space is spanned by the fields and the space , is spanned by , respectively. It is clear to see that when , is Engel group.

Let be the Carnot-Carathéodory distance on the Goursat group. A homeomorphism between domains on the Goursat group is* quasiconformal* if
is uniformly bounded. If in addition the homeomorphism is *-quasiconformal*.

On , a 1-quasiconformal mapping is holomorphic or antiholomorphic. If we identify the Heisenberg group with a quadratic hypersurface in , 1-quasiconformal mapping on the quadratic hypersurface is the restriction of a holomorphic or antiholomorphic mapping on [3]. This can be viewed as the higher dimensional case of the holomorphicity of 1-quasiconformal mapping. This phenomenon happens for Goursat group.

A* CR manifold* is a smooth manifold equipped with a distribution of even rank and a complex structure as endomorphisms with . Denote
A CR manifold is supposed to satisfy the following integrability condition: , if . The number is called* CR dimension of * and is called the* codimension of *. A smooth mapping is called* CR* if
CR manifolds appear naturally as embedded real submanifolds of complex manifolds. The distribution is then defined as and is the restriction of the complex structure in the ambient complex manifold to . We recall that, for a manifold with , the induced CR structure on is given by , and .

The Goursat group can be realized as the real submanifold of : where , , , by the diffeomorphism defined by It is easy to check that where , , , is the complex tangent vector of , and we can define the standard CR structure on by and . We can also define quasiconformal mappings on by (5) with respect to its Carnot-Carathéodory distance defined in Section 2. There is one to one corresponding between 1-quasiconformal mappings on and via the diffeomorphism . Therefore, if we want to study 1-quasiconformal mappings on , we can just study the 1-quasiconformal mappings on . For further details see Section 2.

Denote We show that 1-quasiconformal mappings between domains on Goursat group whose Pansu differentials preserve are CR or anti-CR.

Theorem 1. *Let be a domain in . Suppose that is a 1-quasiconformal mapping with preserving ; then is CR or anti-CR; that is, either
**
or
**
where and is some -differential point.*

*Remark 2. *By this theorem the determination of 1-quasiconformal mappings is reduced to the determination of CR automorphisms of a submanifold in . According to the extension theorem (Theorem 1.4 in [13]), locally, the smooth CR automorphisms of are holomorphic automorphisms of , which is also an interesting problem.

*Remark 3. *In the cases of Heisenberg group and Engel group, the condition of preserving in Theorem 1 can be omitted. However, by a similar method as in those cases, we cannot remove it, since it is inconvenient to use the expressions of Baker-Campbell-Hausdorff formula for groups of higher step.

#### 2. Preliminaries

There is a natural homogeneous norm on , given by and an associated pseudometric , given by for . The metric is left-invariant and is related to the dilations by the formula where denotes the dilation on .

The* horizontal tangent space at * is a subspace of the tangent space , and it is spanned by the vector fields and . An absolutely continuous curve is* horizontal* if its tangent vectors , , lie in the horizontal tangent space . By Chow [14], any given two points can be connected by a horizontal curve.

On , we fix a quadric form with respect to which the vector fields , are orthonormal at every point and satisfy
where . is called* infinitesimal Carnot-Carathéodory (C-C) metric*. Then the* length * of the curve is defined by

The* Carnot-Carathéodory distance (C-C distance)* is the infimum of the lengths of all absolutely continuous horizontal curves connecting and :
According to [15], this metric is equivalent to the pseudometric; that is,
for suitable constants .

For the manifold , the horizontal tangent space at is . On , we fix a quadric form with respect to which , are orthonormal at every point and satisfy where . The C-C distance on can be defined as above.

By definition (8), in corresponding to is defined by the equation where It is direct to check that And the vector field defined by (10) satisfies That is, is a complex tangent vector field of , and is a CR manifold with the induced CR structure . We obtain the following relation between and .

Proposition 4. * is CR diffeomorphic and isometric to via defined by (9); that is, and are CR and isometric.*

*Proof. *We can denote the point in as where , and then the group multiplication of can be written in the following form:
Then the complex horizontal space is spanned by , where
here .

We claim that, under this diffeomorphism defined by (9), the complex tangent vector fields of are mapped onto and , respectively. In fact, for any and ,
Therefore,
Similarly

It is clear to see that is
where

Similarly, we find
Hence, are CR and by (30), (31), and (34) together with (18) and (22), we get for any and for any , so preserve C-C distance. This proves the proposition.

#### 3. -Differential of 1-Quasiconformal Mappings on

The notion of differential of a map between domains in Carnot group is due to Pansu [16]. Let denote left-translation on . The mapping ( are domains in ) is called* p-differentiable at * if, for , the mappings
converge locally uniformly to a homomorphism of the group , which is called* Pansu differential (P-differential)*.

The Pansu differential is a Carnot homomorphism. Denote by the induced Lie algebra homomorphism of the group homomorphism . According to [17], is grading-preserving. As a result of [16] or [18], quasiconformal mappings between domains in Carnot group are -differentiable a.e. The result is valid on the group as it is a Carnot group.

Define the bilinear form on by Then by , the corresponding matrix is

Let be a 1-quasiconformal mapping between domains in ; since is grading-preserving, then maps onto itself, respectively. In particular , so for any , for some matrix . Since , then for some .

Proposition 5. *The matrix above can be written as the following form:
**
for some , where for any with , ; depend on .*

*Proof. *Since , now suppose
then

Because is Lie algebra homomorphism, that is,
together with , we find that
namely,

By Lemma 3.3 in [19], for the 1-quasiconformal mapping , we have
where is defined by (18), that is,
Therefore, there exists , such that
for any , . Then is an orthogonal matrix. Together with (45), we have
that is,
Thus
Therefore, . Note that , where is the unit matrix; hence
From (51), (52) and setting , , we have (40). The result follows.

We get the following relation between -differential and the usual derivatives. For the corresponding result on Heisenberg group see p.31 of [3]. In the following, the letter will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.

Proposition 6. *Suppose is a quasiconformal mapping between domains in with , and if is -differentiable at 0 and preserves the plan in (11), then is differentiable in the usual sense at 0 along the horizontal directions and
**
where and is the P-derivative of at 0.*

*Proof. *Since is -differentiable at , by definition, there exists such that, for and ,
Recall that the homomorphism commutes with ; then for points in particular, we have
Therefore

Now compare this with Euclidean distances. Since preserves the plane, that is,
for some , then
Assume that ; we get
It follows that
Therefore,

From Lemma 3.3 in [19], is Lipschitz; then by the equivalence of and , we have
for some , namely,

Since is continuous, then when , we have
where is a positive constant. Now substituting (64) and (65) into (62) and then by iteration, we obtain
Then by (61), for any , we have

If we let , then
namely,
Therefore,
Similarly
The proposition is proved.

We need the following relation between -differential and the usual tangential mapping. See Proposition 3.2 in [10] for the corresponding result on the Heisenberg group.

Proposition 7. *Let be a quasiconformal mapping between domains in the Goursat group . Suppose is -differentiable at and preserves the plan in (11); then
**
for , where and are the tangential mappings of for .*

*Proof. *Suppose is -differentiable at ; let ; then is also a quasiconformal mapping and . Moreover, by Chain rule (Lemma 3.7 in [19]), is -differentiable at the origin 0, and
Since is the identity automorphism, then
Therefore, preserves the plan .

By Proposition 6, we have
From (2) we can see that and . Therefore,
Consequently,
This completes the proof.

We need the following proposition due to Capogna ([19], Theorem 1.1).

Proposition 8. *Suppose that and are open subsets of Carnot groups and , and is 1-quasiconformal. Then is smooth.*

*Proof of Theorem 1. *Let be a 1-quasiconformal mapping satisfying the hypothesis in Theorem 1. By Proposition 4, defined by (9) and (32) are CR and isometric; then is a 1-quasiconformal mapping between domains in . Suppose is a -differentiable point of ; then is grading-preserving; that is, for some , and . From Proposition 5, at , has form (40), and , where for any with .

In the complexified subspace , the corresponding complex basis is , , where is as (28). On the matrix level
with . Then

We claim that , . In fact, by (28) and (40), we have
Similarly, .

From Proposition 8, is smooth, and therefore is also smooth. We write the 1-quasiconformal mapping in the form
with
where . Then
where

In fact, for any ,

Let ; by (30), (34), (72), and (80) together with the left-invariant property of ,
Since is -differentiable a.e., we have
Consequently, in (83) have the following forms
Hence, if , then , ; if , then , . Since and are smooth, Theorem 1 follows.

*Remark 9. *The CR automorphism of is unknown now. It is also very interesting to determine it, since if this can be solved, we can get the 1-quasiconformal mappings on these groups.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was partially supported by NSF of China (Grant nos. 11271175, 11301248 and 11326079), the DYSP, and AMEP of Linyi University.