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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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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 $\cosh^2(t)$ ? –  Thomas Richard Feb 25 '13 at 8:26
    
Sorry,I should write$M = R \times {}_{\cosh \left( t \right)}N$ –  jiangsaiyin Feb 25 '13 at 9:28
    
It must be a manifold,please help me to close the question if you see it. –  jiangsaiyin Feb 26 '13 at 3:17
    
Since you raised this question, can you please tell us why it must be a manifold? –  YangMills Mar 1 '13 at 2:41
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