What are the transformations that preserve cross ratios on a sphere in higher dimensions? If we have four points $x,y,z,w$ on a sphere, then the cross ratio is $\frac{|x-z|}{|x-w|}\frac{|y-w|}{|y-z|}$.
If we consider $S^1 \subseteq \mathbb{C}$, then the transformations of $\mathbb{C}$ which preserve the cross ratios on the circle are precisely the Mobius transformations which map the open unit disc to itself.
Is there a nice classification for transformations preserving cross ratios of spheres in higher dimensions? 
 A: In any dimension, the Mobius group generated by inversions in spheres preserves cross-ratios, and it also contains all translations and scalings, as well as all linear isometries. In fact, the Mobius group is exactly the group of maps preserving cross-ratios.
The proof of the fact that every map $f:S^n \to S^n$ which preserves cross-ratios is a Mobius transformation proceeds by induction, but it is a little messy. The main idea is that you can always compose $f$ with a Mobius transformation to fix any three points, and then use stereographic projection to throw these points to $0$, $\infty$, and some unit vector $e \in \mathbb{R}^n$. Then preservation of cross ratios shows that this new map preserves the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$, so by induction assumption the restriction to this unit sphere is a Mobius transformation. Since the points to start with are arbitrary, this shows that the restriction of $f$ to any $(n-1)$-dimensional subsphere of $S^n$ is a Mobius transformation. The messy part is to show that this implies that the map $f$ is globally a Mobius transformation. (I believe there must be an elegant way to do this, but I cannot think of one right now.)
