How to calculate the scalar curvature under a change of metric in a Riemannian 3-manifold
Clash Royale CLAN TAG#URR8PPP
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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
add a comment |
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
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
asked Sep 11 at 3:52
陶夕夕
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