A construction of inner maps preserving the Haar measure on spheres

Boguslaw Tomaszewski

Vol. 115 (1984), No. 1, 203–215

DOI: 10.2140/pjm.1984.115.203

Abstract

We show, for n ≥ m, the existence of non-trivial inner maps f : Bⁿ → B^m with boundary values f_∗ : Sⁿ → S^m such that f_∗⁻¹(A) has a positive Haar measure for every Borel subset A of S^m which has a positive Haar measure. Moreover, if n = m, the equality σ(f_∗⁻¹(A)) = σ(A) holds, where σ is the Haar measure of S^m.

Mathematical Subject Classification 2000

Primary: 32A35

Secondary: 32A40

Milestones

Received: 11 February 1983

Published: 1 November 1984