## How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$

$$M = R \times N$$with the warped product metric$$d{s^2} = d{t^2} + {\cosh ^2}\left( t \right)ds_N^2$$where N(dimN=n-1) is a compact manifold with $$Ric \ge - \left( {n - 2} \right)$$It should be mentioned that M may not be a Riemannian manifold but an Alexandrov space.So how to compute the first eigenvalue of M?If we restrict to the case $$N = {S^{n - 1}}\left( {\frac{1}{2}} \right)$$an n-1 dim sphere with radius 1/2,then the result?

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 It would be helpful if you gave a reference for the definition of a hyperbolic cone over a metric space. Presumably you want some kind of warped product with warping factor $\sinh$. – Agol Jan 26 at 23:44