Angles and quasiconformal mappings on Riemannian manifolds. (English) Zbl 0712.58009

The author gives a characterization of quasiconformal mappings between infinite dimensional Riemannian manifolds.
Let \(1\leq K<\infty\). Let f: \(V\to \tilde V\) be a homeomorphism. Then f is said to be K-quasiconformal provided that for all \(x\in V\) and angle \(\alpha\), (1/K)\(\alpha\leq f(\alpha)\leq K\alpha.\)
The following theorem is proved. Let f: \(V\to \tilde V\) be a local diffeomorphism. Then f is K-quasiconformal in V iff \(\| Q_ xf\|^ 2\leq K\) for every \(x\in V\). Here \(Q_ xf=\| (T_ xf)^{- 1}\|^{1/2}\| T_ xf\|^{1/2}T_ xf\).
Reviewer: J.E.Keesling


58B10 Differentiability questions for infinite-dimensional manifolds
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
53C20 Global Riemannian geometry, including pinching