Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
Try $f$ linear with determinant $1$. –  Gerry Myerson Jan 24 '13 at 11:44
    
@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

1 Answer 1

up vote 2 down vote accepted

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. $

share|improve this answer
    
There's an echo in here. –  Gerry Myerson Jan 24 '13 at 11:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.