Let $f(x)$ be the bump function $$ f(x_1,\dots,x_N)\triangleq \prod_{i=1}^N exp\left( \frac{1}{1-x_i^2} \right), $$ $I$ be the identity matrix and, $R$ be a special orthogonal matrix not equal to $-I$. Then is $$ x\mapsto (f(x)R + (1-f(x))I)\cdot x, $$ a $C^1(\mathbb{R}^N,\mathbb{R}^N)$-diffeomorphism (moreover, is it analytic)?

  • $\begingroup$ I assume you mean to define $f$ piecewise, so that it vanishes outside the cube $|x_i|<1$? This will certainly not be analytic (think about the unique continuation), and I doubt it's a diffeomorphism - just the obvious failure in the case of $R = -I$ is far from special. If you want to interpolate between rotations like this then it's a better idea to do your interpolation inside $SO(n)$ rather than linearly. $\endgroup$ – Anthony Carapetis Mar 2 '18 at 2:56
  • $\begingroup$ So you're suggesting something like: $x\mapsto exp(f(x)X)\cdot x$ would work? (Where $X\in so(D)$?) $\endgroup$ – AIM_BLB Mar 2 '18 at 3:19
  • $\begingroup$ Perfect! That works just like I wanted thanks! Unless I'm missing something, I'll post the answer and gie you some points too...once I can reward :) $\endgroup$ – AIM_BLB Mar 2 '18 at 3:24
  • $\begingroup$ Yeah, that's essentially what I meant, though it's only obvious that it works if you use an $f$ that is radially symmetric. $\endgroup$ – Anthony Carapetis Mar 2 '18 at 7:53
  • $\begingroup$ Awesome! Thanks a ton! I'll write it up shortly:) $\endgroup$ – AIM_BLB Mar 2 '18 at 12:26

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