Are isometries the only geodesic preserving maps in a CAT(0)-space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:14:44Z http://mathoverflow.net/feeds/question/15805 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15805/are-isometries-the-only-geodesic-preserving-maps-in-a-cat0-space Are isometries the only geodesic preserving maps in a CAT(0)-space? HenrikRüping 2010-02-19T10:34:19Z 2010-02-20T04:39:41Z <p>Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with that map in the sense, that $s(f(x),f(y),t)=f(s(x,y,t))$. Then one can ask, whether any self-homeomorphism of $X$, which is compatible with $s$ in the upper sense is already a isometry.</p> <p>This is clearly wrong for $X=\mathbb{R}^n$, as all affine maps are compatible with $s$. So the question is, whether these are the only examples.</p> <p>For example I think I can show, that the $n$-dimensional hyperbolic space ($n\ge 2$) is rigid in that sense.</p> <p>EDIT: Due to the big amout of counterexamples one could better ask the following question:</p> <p>Are the spaces $\mathbb{R}^n$ the only spaces, which have self homeomorphisms compatible with $s$ (in the upper sense), that are not self-similarities ?</p> http://mathoverflow.net/questions/15805/are-isometries-the-only-geodesic-preserving-maps-in-a-cat0-space/15814#15814 Answer by Guntram for Are isometries the only geodesic preserving maps in a CAT(0)-space? Guntram 2010-02-19T14:33:21Z 2010-02-19T14:33:21Z <p>If $Y$ is any CAT(0)-space, $X:=R^n \times Y$ will give another counterexample, so you might want to restrict to indecomposable $X$.<br /> The next remark is that the union of the coordinate axes in $R^2$ gives another example, as does an arbitrary union of lines through the origin in $R^n$ with the induced length metric.</p> <p>Another one in the same spirit is provided by the "forest" obtained by attaching a half-line to every point in $R^2$. Maybe it's true that that every self-similar CAT(0)-space $(X,d)$, i.e. one which is isometric to $(X,\rho \cdot d)$ for some $\rho \neq 1$, is a counter-example?</p> http://mathoverflow.net/questions/15805/are-isometries-the-only-geodesic-preserving-maps-in-a-cat0-space/15822#15822 Answer by Anton Petrunin for Are isometries the only geodesic preserving maps in a CAT(0)-space? Anton Petrunin 2010-02-19T16:45:37Z 2010-02-20T04:39:41Z <p>The map which you call "geodesic preserving" is usually called "affine". It seems that affine maps to the real line are well understood even for general length space.</p> <p>For your later edit: you may always take two spaces which admit self-similar maps and consider map on the product which move each coordinate with different coefficients.</p> <p>A. Lytchak says that the following is a well known open question:</p> <blockquote> <p>If such map exist then the space can be embedded into product of spaces and the map preserves product structure.</p> </blockquote> <p>P.S. The "example" I gave before was not an example.</p>