How to calculate the scalar curvature under a change of metric in a Riemannian 3-manifold

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Let $g=text d t^2+ gamma_t$ be the metric of a Riemann manifold $M$ of dimension 3 , where $gamma_t$ is the metric on the surface $Sigma_t=partial M$.



If $hatg=rho^2(t)text dt^2+gamma_t$



then the scalar curvature $hatR$ of $hatg$ is



$hatR(t,x)=frac1rho^2(t)(R(t,x)+2K(t,x)(rho^2(t)-1)+frac2rho'(t)rho(t)H(t,x))$



where $K(t,x)$ and $H(t,x)$ are the Gauss and mean curvature of $Sigma_t$ with respect to $g$ respectively.



Here I don't know how to calculate $hatR$ ,I need someone to help me .






geometry curvature







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    Let $g=text d t^2+ gamma_t$ be the metric of a Riemann manifold $M$ of dimension 3 , where $gamma_t$ is the metric on the surface $Sigma_t=partial M$.



    If $hatg=rho^2(t)text dt^2+gamma_t$



    then the scalar curvature $hatR$ of $hatg$ is



    $hatR(t,x)=frac1rho^2(t)(R(t,x)+2K(t,x)(rho^2(t)-1)+frac2rho'(t)rho(t)H(t,x))$



    where $K(t,x)$ and $H(t,x)$ are the Gauss and mean curvature of $Sigma_t$ with respect to $g$ respectively.



    Here I don't know how to calculate $hatR$ ,I need someone to help me .






    geometry curvature







    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $g=text d t^2+ gamma_t$ be the metric of a Riemann manifold $M$ of dimension 3 , where $gamma_t$ is the metric on the surface $Sigma_t=partial M$.



      If $hatg=rho^2(t)text dt^2+gamma_t$



      then the scalar curvature $hatR$ of $hatg$ is



      $hatR(t,x)=frac1rho^2(t)(R(t,x)+2K(t,x)(rho^2(t)-1)+frac2rho'(t)rho(t)H(t,x))$



      where $K(t,x)$ and $H(t,x)$ are the Gauss and mean curvature of $Sigma_t$ with respect to $g$ respectively.



      Here I don't know how to calculate $hatR$ ,I need someone to help me .






      geometry curvature







      share|cite|improve this question













      Let $g=text d t^2+ gamma_t$ be the metric of a Riemann manifold $M$ of dimension 3 , where $gamma_t$ is the metric on the surface $Sigma_t=partial M$.



      If $hatg=rho^2(t)text dt^2+gamma_t$



      then the scalar curvature $hatR$ of $hatg$ is



      $hatR(t,x)=frac1rho^2(t)(R(t,x)+2K(t,x)(rho^2(t)-1)+frac2rho'(t)rho(t)H(t,x))$



      where $K(t,x)$ and $H(t,x)$ are the Gauss and mean curvature of $Sigma_t$ with respect to $g$ respectively.



      Here I don't know how to calculate $hatR$ ,I need someone to help me .






      geometry curvature




      geometry curvature






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      asked Sep 11 at 3:52









      陶夕夕

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