M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie $M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov space,must M be a manifold?must N be totally geodesic in M?If not,please give counterexamples and point out the singular points.
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3$\begingroup$ I don't understand what can go wrong, the metric you write is just a smooth warped product metric on $N\times\mathbb{R}$. Is the warping factor really $\cosh^2(t)$ ? $\endgroup$– Thomas RichardCommented Feb 25, 2013 at 8:26
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$\begingroup$ Sorry,I should write$M = R \times {}_{\cosh \left( t \right)}N$ $\endgroup$– jiangsaiyinCommented Feb 25, 2013 at 9:28
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$\begingroup$ It must be a manifold,please help me to close the question if you see it. $\endgroup$– jiangsaiyinCommented Feb 26, 2013 at 3:17
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$\begingroup$ Since you raised this question, can you please tell us why it must be a manifold? $\endgroup$– YangMillsCommented Mar 1, 2013 at 2:41
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