We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$ , for n ≥ 2, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then f is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then f is η-quasisymmetric in the hyperbolic metric, where η depends only on n and K. We obtain the same result for hyperbolic n-manifolds. Analogous results in $\mathbb{R}^n$ , and metric spaces that behave like $\mathbb{R}^n$ , are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$ .

中文翻译：

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准共形性和双曲偏斜

我们证明如果$f:\mathbb{B}^n \to \mathbb{B}^n$， 为了n≥ 2，是在所有等边双曲三角形上具有有界偏斜的同胚，则F实际上是准共形的。相反，我们证明如果$f:\mathbb{B}^n \to \mathbb{B}^n$那么是准共形的F是η-双曲度量中的准对称，其中η只取决于n和ķ. 我们对双曲线得到相同的结果n-歧管。类似的结果$\mathbb{R}^n$, 以及行为类似的度量空间$\mathbb{R}^n$, 是已知的, 但据我们所知, 这些是双曲线设置中的第一个这样的结果, 这是使用的自然度量$\mathbb{B}^n$.