Bounding the volume distortion of a manifold

[eyenir]

Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$.

Its volume distortion is defined as $G=det(D_f^tD_f).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$.

What happens for $n>1$? Can one bound from below this $G$? If so: under which assumptions?

 

[wrobel]

Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection.

did not you know that the image of a compact set under a continuous mapping is a compact set?

 

[eyenir]

So you suggest taking a finite sub-cover ${U_i}$ such that $M|_{U_i}=\{(x_1,\dots,x_{n-1},g_i\}$ where $g_i:\mathbb{R^{n-1}\to R}$ is a $C^1$ function on $U_i$?