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Consider $\mathbb{R}^n$ with the standard volume form $\omega_{0}:=dx_1\wedge ...\wedge dx_n$ . Let $r$ and $R$ be two positive real numbers, with $R > r$ . Show that for every $a, b \in B (0, r )$, there exists a diffeomorphism $f$ of $\mathbb{R}^n$ such that $f(a) = b , f(x)= x$ for $|| x || > R$ and $f ^{*}\omega_{0} = \omega_{0}$.

One-dimensional case is fine. I have tried to extend that idea to the higher dimensional case. I have an exam. I don't understand how to construct $f$. Hints are welcomed. Can anyone please help me?

Souvik Dey
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Asad13
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1 Answers1

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$\newcommand{\Reals}{\mathbf{R}}$Hint: Let $(r, \theta)$ denote polar coordinates on $\Reals^{2}$. If $\phi:(0, \infty) \to \Reals$ is an arbitrary smooth function, the formula $$ f(r, \theta) = \bigl(r, \theta + \phi(r)\bigr) $$ defines a "rotational shear" mapping that is smooth and area-preserving.

Andrew D. Hwang
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  • If $b=(r_1,\phi), a=(r,\phi_1)$, then $f(a) \neq b$,as $f$ preserves the distance from origin.Then how should I use this function? – Asad13 Apr 29 '17 at 17:42
  • You might put the origin of your polar coordinate system at the midpoint $O = \frac{1}{2}(a + b)$, for example. – Andrew D. Hwang Apr 29 '17 at 17:56