A continuous area-preserving mapping is an isometry?

Question

Suppose that $f\colon\mathbb R^2\to\mathbb R^2$ is a continuous map which preserves area in the Euclidean sense. Can we say that $f$ is an isometry?

Note. We donot assume that $f$ is differentiable.

@Avatar, the MO question is interesting, but not directly relevant, is it? — Gerry Myerson, Jan 24 '13 at 11:55
@GerryMyerson: i understood your point,sorry; i am deleting my comment. — Aang, Jan 24 '13 at 12:11

score 3 · Accepted Answer · answered Jan 24 '13 at 11:52

3

consider the matrix $$A = \left( \begin{array}{ccc} 2 &0 \\ 0 & 1/2 \\ \end{array} \right) $$ and the continuous function $y=Ax$.

The area of any set remains the same under this linear transformation, but $|A \hat{x}| = 2 > 1. $

answered Jan 24 '13 at 11:52

guest196883

1 Answers1