# Squared displacement function on Lorentzian manifolds

Hi,
Let $\varphi$ be an isometry of a simply connected pseudo Riemannian manifold $M$. The squared displacement function of $\varphi$ is $d^2_{\varphi}(p):=d^2(\varphi (p),p)$, $p\in M$, where $d$ is the distance on $M$. By a theorem of J.A.Wolf, if $M$ is complete Riemannian manifold of negative sectional curvature then $\varphi$ is the identity iff $d^2_{\varphi}$ is constant (and $d^2_{\varphi}$ is constant iff it is bounded).
Now, my question is that if $\varphi$ is an isometry of a Lorentzian space form (specially $\widetilde{AdS}_n$, the universal covering of the anti de Sitter space) is there any sufficient condition under which one can deduce that $\varphi$ is the identity?
Thanks.

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Does the Lorentzian space form have constant zero curvature? In which case, it looks like you're looking for some kind of generalization of Wolf's result to manifolds of nonpositive sectional curvature. Do you have a reference for Wolf's theorem? –  Oliver Jones Jun 15 '13 at 0:59
The original reference of Wolf's theorem is: WOLF, J.A.: Homogeneity and bounded isometries in manifolds of negative curvature. Ill. J. Math 8 (1964), 14-18. –  purelymath Jun 23 '13 at 8:49