How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:27:08Z http://mathoverflow.net/feeds/question/119831 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119831/how-to-compute-the-first-eigenvalue-of-m-r-times-cosh-tn How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$ jiangsaiyin 2013-01-25T13:40:09Z 2013-01-27T06:03:12Z <p>$$M = R \times N$$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?</p>