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It is well-known that the ends of a finite-volume hyperbolic manifold are warped products $$(0,\infty)\times_f T$$ for some euclidean manifold $T$ and $f(t)=e^{-t}$.

Question: Is there a similar result for rank 1 symmetric spaces (or even for more general negatively curved manifolds)?

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It may be not the answer you are hoping for but still: one can show (short tensor calculations) that, even locally, the metric of an irreducible symmetric space different from the space of constant curvature can not be a warped product metric.

Now, concerning arbitrary negatively curved metrics: I do not really understand what you would like to know; let me give three answers on different versions of your question and if possibly please follow the rules and write the motivation for your question

(1) of course a generic metric, even negatively curved, is not a warped product metrics.

(2) Of course there exist examples of negatively curved metrics such that they are warped products near cusps -- just take a manifold of constant negative curvature and slightly perturb it locally, ``far from cusp''.

(3) Warped structure relates to the existence of a nontrivial solution for the equation $$ \textrm{Trace-free-part-of}(\nabla \nabla u)=0 \ \ \ (\ast)$$ which is a system of linear overdetermined PDEs on the unknown function $u$. In a neighborhood of almost every point, metrics admitting a solution of $(\ast)$ are locally warped product, and warped product metrics locally admit a solution of such equation.

On can show that complete negavely curved manifolds of finite volume do not have nontrivial solutions of $(\ast)$.

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  • $\begingroup$ Do you have a reference for point (3)? I am having a bit of difficulty verifying it myself, and would love to see it done. Thanks. $\endgroup$ Nov 19, 2013 at 14:19
  • $\begingroup$ Ah, you are talking about Cheeger-Colding, and you restrict only to consider warped products of the form $(a,b) \times_f N$! Okay, problem solved. Sorry for the noise. (For those who care, point (3) is given in the first section of jstor.org/stable/2118589 ) $\endgroup$ Nov 19, 2013 at 14:38
  • $\begingroup$ Willi, I did not know about Cheeger-Colding (thanks for the reference). My proof, which actually can be generalised for other (not only $R\times_f M$) warped products is as follows: the existence of a solution of (*) allows one to construct an integral for the geodesic flow of the metric, which must be trivial on the manifolds under consideration because the geodesic flow is ergodic. $\endgroup$ Nov 19, 2013 at 17:06

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