1 Introduction and Main Result

It is known that there are many equivalent ways of defining quasiconformal mappings in Euclidean spaces and even in metric spaces, e.g. via the conformal moduli of quadrilaterals or rings, the extremal lengths, the distortion of infinitesimal spheres or balls or equilateral triangles, see [3, 10,11,12] and the references therein. These definitions play important roles in dealing with many research problems [1, 2, 12, 15] and are also useful in different situations, see [5, 10, 11, 13].

In 1965, Agard and Gehring [2] investigated the distortion of angles under planar quasiconformal mappings in order to generalize the fact that a conformal mapping is angle-preserving. For this reasoning, they introduced the notions of topological angles and inner angular measures between intersecting arcs, and they obtained some angular characteristics for the class of quasiconformal mappings in the plane, see [2, Theorem 4]. Later, Agard demonstrated some equivalent conditions for quasiconformality in terms of topological angles and their measures in higher dimensional Euclidean spaces, see [1, Theorems 5.1 and 5.2].

In 2005, Aseev et al. in [4] generalized the idea of Agard and Gehring by introducing the following definition of an angle between sets in metric spaces.

Definition 1

Let \(A_1\) and \(A_2\) be two nonempty subsets of a metric space X with \(A_1\cap A_2\ne \emptyset \) and \({\text {diam}}(A_1\cup A_2)>0\). The angle between \(A_1\) and \(A_2\) is defined to be

$$\begin{aligned} \angle (A_1,A_2)=\inf _{x_1\in A_1,x_2\in A_2}\sup _{z\in A}\frac{|x_1-x_2|}{|x_1-z|+|x_2-z|},\end{aligned}$$
(1)

where \(A=A_1\cap A_2\).

Remark 1

If \({\text {diam}}(A_1\cup A_2)=0\), then \(A_1=A_2=\{x\}\) for some \(x\in X\). Thus the inequality (1) is true for all \(c>0\), which leads to \(\angle (A_1,A_2)=\infty \). Hence, to exclude this possibility, we assume that \({\text {diam}}(A_1\cup A_2)>0\), i.e., \(A_1\cup A_2\) contains at least two points. Note that \(0\le \angle (A_1,A_2)\le 1\). Next we give some examples.

  1. (1)

    If \(A_1=A_2=[0,1]\subset {\mathbb R}\), then \(\angle (A_1,A_2)=0\).

  2. (2)

    If \(A_1=A_2=\{x,y\}\), then \(\angle (A_1,A_2)=1\).

  3. (3)

    If \(A_1=A_2=\{1,2,3\}\subset {\mathbb R}\), then \(\angle (A_1,A_2)=1/3\).

  4. (4)

    If \(A_1=\mathbb {Q}\cap [0,1]\) and \(A_2=([0,1]{\setminus } A_1)\cup \{0,1\}\), then \(\angle (A_1,A_2)=0\).

In this paper, X and Y are assumed to be metric spaces. The polish notation \(|x-y|\) is used for the distance between x and y in any metric space. The primes always stand for the images of points and sets under a mapping f. For example \(x'=f(x)\) and \(A'=f(A)\). Other notions and notations in this section will be explained in Sect. 2.

In [4], the authors observed that the notion of metric angles can be used to characterize the class of quasisymmetric mappings.

Theorem 1

( [4, Theorem 2.1]) Suppose that \(f: \, X\rightarrow Y\) is a homeomorphism. Then f is \(\eta \)-quasisymmetric if and only if there exist continuous strictly increasing functions \(\varphi \) and \(\psi \) on [0, 1] such that \(\varphi (0)=0\) and \(\psi (0)=0\) and the inequalities \(\angle (A_1', A_2')\ge \psi (\angle (A_1, A_2))\) and \(\angle (A_1, A_2)\ge \varphi (\angle (A_1', A_2'))\) hold for all subsets \(A_1, A_2\subset X\).

On the one hand, the positive angle condition is helpful to study the gluing of quasisymmetric mappings and quasimöbius mappings, see e.g. [4, 7, 8]. On the other hand, the procedures of sewing quasisymmetric mappings and quasimöbius mappings have recently found applications in the study of hyperbolic groups with planar boundaries and the planar Schönflies theorem for bilipschitz and quasisymmetric mappings, see [9, 14, 15].

The above background motivates us to consider the following question:

Question 2

Is there an angular characteristic for the class of quasimöbius mappings in \(\mathbb {R}^n\) or metric spaces?

As a special class of quasimöbius mappings, the inversion transformation is obviously quasimöbius but not quasisymmetric. Therefore, one finds from Theorem 1 that a quasimöbius mapping does not preserve positive angles unless certain additional assumptions are imposed. In this paper, a concrete example will be presented to explain this phenomenon, see Example 1 in Sect. 4.

As our main result, we give an affirmative answer to Question 2 as follows.

Theorem 2

Suppose that \(f: \, X\rightarrow Y\) is a homeomorphism between metric spaces. Then f is \(\theta \)-quasimöbius for some homeomorphism \(\theta : [0,\infty )\rightarrow [0,\infty )\) with \(\theta (0)=0\) if and only if there is some continuous, strictly increasing function \(\varphi : \,[0, 1]\rightarrow [0,1]\) with \(\varphi (0)=0\) with the following property: for all subsets \(A_1, A_2\subset X\) satisfying

$$\begin{aligned} \angle (A_1, A_2)\ge t \;\;\;\;\text{ and }\;\;\;\;{\text {diam}}(A_1\cap A_2)\ge s\min \{{\text {diam}}(A_1), {\text {diam}}(A_2)\}, \end{aligned}$$

for some s, \(t>0\), we have

$$\begin{aligned} \angle (A_1', A_2')\ge \varphi (ts). \end{aligned}$$

\(\varphi \) and \(\theta \) depend only on each other.

Remark 2

The necessity of Theorem 2 can be regarded as a generalization of [7, Lemma 3.2(i)] at two aspects. On the one hand, we do not need the intersection of two given sets to be uniformly perfect. On the other hand, the distortion of angles under metric spaces inversions, introduced by Buckley et al. [6], was investigated by Guan et al. in [7], and what we considered here are quasimöbius mappings. Indeed, our proof is quite different from theirs.

Moreover, one observes from Theorem 2 that the requirements (1) and (4) of [7, Theorem 1.1] is reasonable and in a sense necessary.

The rest of this paper is organized as follows. In Sect. 2, we recall some necessary terminology and several useful known results. Section 3 is devoted to the proof of Theorem 2. Finally, we show the properties of Example 1 in Sect. 4 regarding positive angles and quasimöbius mappings.

2 Preliminaries and Auxiliary Results

Let X be a metric space. The sets A, \(A_1\) and \(A_2\) stand for subsets of X. For a bounded set A, \({\text {diam}}(A)\) means its diameter, and \(x, y,\cdots \) mean the points in X. Given two real numbers s and t, we let

$$\begin{aligned} s\vee t=\max \{s, t\}\,\,\,\text{ and }\,\,\,s\wedge t=\min \{s, t\}. \end{aligned}$$

Next, we recall the definition of quasimöbius mappings introduced by Väisälä in [16]. We refer to [5, 6, 9, 17] for more investigations and applications.

Definition 2

Suppose that \(\theta : \,[0, \infty )\rightarrow [0,\infty )\) is a homeomorphism with \(\theta (0)=0\). A homeomorphism \(f: \, X\rightarrow Y\) is called \(\theta \)-quasimöbius if \(r(a, b, c, d)\le t\) implies

$$\begin{aligned} r(a', b', c', d')\le \theta (t) \end{aligned}$$

for any quadruple of distinct points (a, b, c, d) in X and any number \(t\ge 0\), where

$$\begin{aligned} r(a, b, c, d)=\frac{|a-c||b-d|}{|a-b||c-d|} \end{aligned}$$

denotes the cross ratio of (a, b, c, d).

In [5], Bonk and Kleiner introduced the following useful notation of a quadruple of distinct points (a, b, c, d):

$$\begin{aligned} <a, b, c, d>=\frac{|a-c|\wedge |b-d|}{|a-b|\wedge |c-d|}. \end{aligned}$$

Obviously, if we change the positions of b and c, then we obtain

$$\begin{aligned} r(a, b, c, d)=\frac{1}{r(a, c, b, d)}\,\,\,\text{ and }\,\,\,<a, b, c, d>=\frac{1}{<a, c, b, d>}. \end{aligned}$$

The relation between r(a, b, c, d) and \(<a, b, c, d>\) states as follows.

Lemma 3

( [5, Lemma 2.3]) Let \(\eta _0(t)=3(t\vee \sqrt{t})\) for \(t\ge 0\). Then for every quadruple of distinct points (a, b, c, d) in a metric space X we have

$$\begin{aligned} <a, b, c, d>\le \eta _0(r(a, b, c, d)). \end{aligned}$$

If we change the positions of b and c, then we get

$$\begin{aligned} <a, b, c, d>\ge \eta _1(r(a, b, c, d)) \end{aligned}$$

with \(\eta _1(t)=\frac{1}{\eta _0(1/t)}\), see [5, Page 133] or [18, Lemma 2.1]. By Lemma 3, a direct computation gives the following equivalent condition for quasimöbius mappings.

Lemma 4

A homeomorphism \(f: \, X\rightarrow Y\) is \(\theta \)-quasimöbius for some homeomorphism \(\theta : [0,\infty )\rightarrow [0,\infty )\) with \(\theta (0)=0\) if and only if there exists a homeomorphism \(\theta _1: [0, \infty )\rightarrow [0, \infty )\) with \(\theta _1(0)=0\) such that \(<a, b, c, b>\le t\) implies

$$\begin{aligned} <a', b', c', d'>\le \theta _1(t) \end{aligned}$$

for any quadruple of distinct points (a, b, c, d) in X and any number \(t\ge 0\), where \(\theta \) and \(\theta _1\) depend on each other.

3 Proof of Theorem 2

In this section, we assume that \(f: \, X\rightarrow Y\) is a homeomorphism between metric spaces.

The proof of Theorem 2 is divided into two parts. We first show the necessity which is stated in the following lemma.

Lemma 5

Let \(\theta : [0,\infty )\rightarrow [0,\infty )\) be a homeomorphism with \(\theta (0)=0\). If f is \(\theta \)-quasimöbius, then there is a continuous, strictly increasing function \(\varphi : \,[0, 1]\rightarrow [0,1]\) with \(\varphi (0)=0\) such that for all subsets \(A_1, A_2\subset X\) satisfying

$$\begin{aligned} \angle (A_1, A_2)\ge t>0 \;\;\;\;\text{ and }\;\;\;\;{\text {diam}}(A_1\cap A_2)\ge s({\text {diam}}(A_1)\wedge {\text {diam}}(A_2)), \end{aligned}$$

we have

$$\begin{aligned} \angle (A_1', A_2')\ge \varphi (ts), \end{aligned}$$
(6)

where \(\varphi \) depends only on \(\theta \).

Proof

We only need to show that

$$\begin{aligned} \angle (A_1, A_2)\ge t>0 \;\;\;\;\text{ and }\;\;\;\;{\text {diam}}(A_1\cap A_2)\ge s({\text {diam}}(A_1)\wedge {\text {diam}}(A_2)) \end{aligned}$$

imply that

$$\begin{aligned} \frac{|x_1'-x_2'|}{\inf _{z'\in A} \{|x_1'-z'|+|z'-x_2'|\}}\ge \varphi (ts), \end{aligned}$$
(7)

for all \(x_1\in A_1\) and \(x_2\in A_2\) with \(x_1\ne x_2\), where \(A=A_1\cap A_2\) and

$$\begin{aligned} \varphi (ts)=\frac{1}{2\theta _1(\frac{3}{st})+1}. \end{aligned}$$

Note that the function \(\theta _1\) is from Lemma 4 which depends only on \(\theta \).

If either \(x_1\in A\) or \(x_2\in A\), then it is easy to get

$$\begin{aligned} \frac{|x_1'-x_2'|}{\inf _{z'\in A'} \{|x_1'-z'|+|z'-x_2'|\}}\ge \frac{|x_1'-x_2'|}{|x_1'-x_2'|}=1, \end{aligned}$$

which implies (7).

Thus, it is left to consider the case that \(x_1\in A_1\setminus A\) and \(x_2\in A_2\setminus A\). Without loss of generality, we also assume that \({\text {diam}}(A_1)\le {\text {diam}}(A_2)\). Since \(\angle (A_1, A_2)\ge t\), there exists a point \(x\in A\) for which

$$\begin{aligned} |x_1-x_2|\ge {\frac{t}{2}}(|x_1-x|\vee |x_2-x|). \end{aligned}$$
(8)

By the assumption \({\text {diam}}(A)\ge s{\text {diam}}(A_1)\), there is a point \(y\in A\) such that

$$\begin{aligned} |x-y|\ge \frac{1}{3}{\text {diam}}(A)\ge \frac{s}{3}|x_1-x|. \end{aligned}$$

This, together with (8) and the fact that \(t, s\in [0, 1]\), shows that

$$\begin{aligned} <x_1, x, x_2, y>=\frac{|x_1-x_2|\wedge |x-y|}{|x_1-x|\wedge |x_2-y|}\ge {\frac{t}{2}}\wedge \frac{s}{3}\ge \frac{st}{3}. \end{aligned}$$

Then we have

$$\begin{aligned}<x_1, x_2, x, y>=\frac{1}{<x_1, x, x_2, y>}\le \frac{3}{st}. \end{aligned}$$

Since f is \(\theta \)-quasimöbius, it follows from Lemma 4 that

$$\begin{aligned} <x_1', x_2', x', y'>\le \theta _1{\left( \frac{3}{st}\right) }, \end{aligned}$$

where the function \(\theta _1\) depends only on \(\theta \).

Thus we get

$$\begin{aligned}<x_1', x', x_2', y'>=\frac{1}{{<x_1', x_2', x', y'>}}\ge \frac{1}{\theta _1(\frac{3}{st})}, \end{aligned}$$

which guarantees that

$$\begin{aligned} \frac{|x_1'-x_2'|}{|x_1'-x'|}\vee \frac{|x_1'-x_2'|}{|x_2'-y'|}\ge <x_1', x', x_2', y'>\ge \frac{1}{\theta _1(\frac{3}{st})}. \end{aligned}$$

Therefore, we have

$$\begin{aligned}{} & {} \frac{|x_1'-x_2'|}{\inf _{z'\in A'}\{|x_1'-z'|+|z'-x_2'|\}} \\ \nonumber\ge & {} \left( \frac{|x_1'-x_2'|}{|x_1'-x'|+|x'-x_2'|}\right) \vee \left( \frac{|x_1'-x_2'|}{|x_1'-y'|+|y'-x_2'|}\right) \\ \nonumber\ge & {} \left( \frac{|x_1'-x_2'|}{2|x_1'-x'|+|x_1'-x_2'|}\right) \vee \left( \frac{|x_1'-x_2'|}{2|y'-x_2'|+|x_1'-x_2'|}\right) \\ \nonumber= & {} \frac{|x_1'-x_2'|}{2(|x_1'-x'| \wedge |x_2'-y'|)+ |x_1'-x_2'|} \\ \nonumber\ge & {} \frac{1}{2\theta _1(\frac{3}{ts})+1}. \end{aligned}$$

Let

$$\begin{aligned} \varphi (ts)={\left\{ \begin{array}{ll} \displaystyle \quad \quad \ 0 &{} \text{ for } ts=0,\\ \; \big (2\theta _1(\frac{3}{ts})+1\big )^{-1} &{} \;\text{ for } ts\in (0,+\infty ). \end{array}\right. } \end{aligned}$$

Hence we obtain (7) and then \(\angle (A_1', A_2')\ge \varphi (ts)\). The lemma follows. \(\square \)

Next, we prove the sufficiency.

Lemma 9

Suppose \(\varphi : \,[0, 1]\rightarrow [0,1]\) is a continuous, strictly increasing function with \(\varphi (0)=0\). If for all subsets \(A_1, A_2\subset X\) satisfying

$$\begin{aligned} \angle (A_1, A_2)\ge t>0 \;\;\;\;\text{ and }\;\;\;\;{\text {diam}}(A_1\cap A_2)\ge s({\text {diam}}(A_1)\wedge {\text {diam}}(A_2)), \end{aligned}$$

we have

$$\begin{aligned} \angle (A_1', A_2')\ge \varphi (ts), \end{aligned}$$

then f is \(\theta \)-quasimöbius for a homeomorphism \(\theta : [0,\infty )\rightarrow [0,\infty )\) with \(\theta (0)=0\) depending only on \(\varphi \).

Proof

Choose x, \(x_1\), y, \(x_2\in X\) arbitrarily. Let

$$\begin{aligned} t=<x, x_1, y, x_2>=\frac{|x-y|\wedge |x_1-x_2|}{|x-x_1|\wedge |x_2-y|}. \end{aligned}$$
(10)

Without loss of generality, we may assume that

$$\begin{aligned} |x'-y'|\ge |x_1'-x_2'|. \end{aligned}$$
(11)

Let \(A_1=\{x_1, x, y\}\) and \(A_2=\{x_2, x, y\}\). Then we get \(A=A_1\cap A_2=\{x, y\}\). It follows from (10) that

$$\begin{aligned} |x-y|\ge t(|x-x_1|\wedge |x_2-y|). \end{aligned}$$

An elementary computation shows that

$$\begin{aligned} (t+1)|x-y|\ge t\big ((|x_1-x|+|x-y|)\wedge (|x_2-y|+|x-y|)\big ), \end{aligned}$$

which leads to

$$\begin{aligned} (t+1){\text {diam}}(A)\ge t({\text {diam}}(A_1)\wedge {\text {diam}}(A_2)). \end{aligned}$$

Thus we have

$$\begin{aligned} {\text {diam}}(A)\ge \frac{t}{t+1}({\text {diam}}(A_1)\wedge {\text {diam}}(A_2)). \end{aligned}$$
(12)

Moreover, by (10), we obtain

$$\begin{aligned}{} & {} \big (|x_1-x|+|x-x_2|\big )\wedge \big (|x_1-y|+|y-x_2|\big ) \nonumber \\ {}{} & {} \quad \le \big (2|x_1-x|+|x_1-x_2|\big )\wedge \big (2|x_2-y|+|x_1-x_2|\big ) \nonumber \\ {}{} & {} \quad = 2(|x_1-x|\wedge |x_2-y|)+|x_1-x_2| \nonumber \\ {}{} & {} \quad = \frac{2}{t}(|x-y|\wedge |x_1-x_2|)+|x_1-x_2| \nonumber \\ {}{} & {} \quad \le \frac{2+t}{t}|x_1-x_2| \end{aligned}$$

which ensures that

$$\begin{aligned} \angle (A_1, A_2)=\frac{|x_1-x_2|}{|x_1-x|+|x-x_2|}\vee \frac{|x_1-x_2|}{|x_1-y|+|y-x_2|}\ge \frac{t}{t+2} \end{aligned}$$

and therefore, by the assumption and (12), we have

$$\begin{aligned} \angle (A_1', A_2')\ge \varphi \left( \frac{t^2}{(t+1)(t+2)}\right) . \end{aligned}$$
(13)

Since

$$\begin{aligned} \angle (A_1', A_2')=\frac{|x_1'-x_2'|}{|x_1'-x'|+|x'-x_2'|}\vee \frac{|x_1'-x_2'|}{|x_1'-y'|+|y'-x_2'|}, \end{aligned}$$

it follows from (11) that

$$\begin{aligned} \angle (A_1', A_2')\le \frac{|x_1'-x_2'|}{|x_1'-x'|\wedge |x_2'-y'|}=\frac{|x_1'-x_2'|\wedge |x'-y'|}{|x_1'-x'|\wedge |x_2'-y'|}=<x',x_1', y',x_2'>. \end{aligned}$$

This, together with (13), ensures that

$$\begin{aligned} T=<x',x_1', y',x_2'>\ge \varphi \left( \frac{t^2}{(t+1)(t+2)}\right) \ge \varphi \left( \frac{t^2}{(t+2)^2}\right) , \end{aligned}$$

which guarantees that

$$\begin{aligned} t\le \psi ^{-1}\circ \varphi ^{-1}(T), \end{aligned}$$

where

$$\begin{aligned} \psi ^{-1}(u)=\frac{2\sqrt{u}}{1-\sqrt{u}}. \end{aligned}$$

Note that \(T\rightarrow 0\) implies that \(t\rightarrow 0\).

Now, Lemma 4 asserts that \(f^{-1}\) is \(\theta _1\)-quasimöbius with \(\theta _1\) depending only on \(\varphi \). Since the inverse of a quasimöbius mapping is also quasimöbius (cf. [16]), we know that f is quasimöbius as well. \(\square \)

Proof of Theorem 2

Theorem 2 follows from Lemmas 5 and 9. \(\square \)

4 An Example

The following example shows that a quasimöbius mapping does not preserve positive angles. Let \(\mathbb {C}\) be the complex plane and let \(z=t_1+it_2\) be a point in \(\mathbb {C}\).

Example 1

Consider the inversion transformation \(u: \mathbb {C}{\setminus }\{0\}\rightarrow \mathbb {C}{\setminus }\{0\}\) with

$$\begin{aligned} u(z)=\frac{z}{|z|^2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{ for }\;\;z\in \mathbb {C}\setminus \{0\}. \end{aligned}$$

Let \(A_1=\{n\}_{n=1}^\infty \) and \(A_2=\{1\}\cup \{-n\}_{n=1}^\infty \). Then we have the following:

  1. (1)

    u(z) is \(\theta \)-quasimöbius with \(\theta (t)=t\);

  2. (2)

    \(\angle (A_1, A_2)=1\);

  3. (3)

    \(\angle (u(A_1), u(A_2))=0\).

Proof

Obviously, u is a Möbius mapping and so (1) holds true. Part (2) follows easily from that \(|a_1-a_2|=|a_1-1|+|a_2-1|\) for all \(a_1\in A_1\) and \(a_2\in A_2\) as \(a_2\le 1\le a_1\).

For (3), note that \(u\left( A_1\right) =\left\{ \frac{1}{n}\right\} _{n=1}^{\infty }\) and \(u\left( A_2\right) =\{1\}\cup \{-\frac{1}{n}\}_{n=1}^\infty \). Thus, we have

$$\begin{aligned} \angle (u(A_1), u(A_2))=\lim _{n\rightarrow \infty }\frac{\frac{2}{n}}{|1-\frac{1}{n}|+|1+\frac{1}{n}|} = 0, \end{aligned}$$

which implies the statement (3). \(\square \)