Are isometries the only geodesic preserving maps in a CAT(0)-space? - MathOverflow

Are isometries the only geodesic preserving maps in a CAT(0)-space?

2010-02-19T10:34:19Z

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.

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.

For example I think I can show, that the $n$-dimensional hyperbolic space ($n\ge 2$) is rigid in that sense.

EDIT: Due to the big amout of counterexamples one could better ask the following question:

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 ?

Answer by Guntram for Are isometries the only geodesic preserving maps in a CAT(0)-space?

2010-02-19T14:33:21Z

If $Y$ is any CAT(0)-space, $X:=R^n \times Y$ will give another counterexample, so you might want to restrict to indecomposable $X$.
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.

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?

Answer by Anton Petrunin for Are isometries the only geodesic preserving maps in a CAT(0)-space?

2010-02-19T16:45:37Z

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.

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.

A. Lytchak says that the following is a well known open question:

If such map exist then the space can be embedded into product of spaces and the map preserves product structure.

P.S. The "example" I gave before was not an example.