Mixing
Ricci Tensor and Einstein Manifolds
$begingroup$
What can we say about an hypersurface Einstein manifolds on $mathbb{R}^{n+1}$ when $ngeq 3$ ?
The manifolds is Einstein so Ricci tensor is a multiply of the metric $g$ on manifold:
$Ric=lambda g$ where $lambdain C^infty(M)$
If $S$ is the scalar curvature of $M$ then the relationship between $S$ and $lambda$ is
$lambda=frac{S}{n}$
Because the scalar curvature of $M$ is the trace of Ricci tensor.
So using the condition that $ngeq 3$ and using a contraction of the second Bianchi Identity we have that the scalar curvature of $M$ is costant so $lambda$ is costant.
Another condition that we have is that $M$ is an hypersurface of $mathbb{R}^{n+1}$ so we can use also the Codazzi-Mainardi Equation ti get some equation between the mean curvature $H$ of $M$, the metric $g$ and the second fondamental formula of $M$.
What can other we say about $M$, it is possible classify it?
general-topology geometry analysis riemannian-geometry riemann-surfaces
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
$endgroup$
add a comment |
$begingroup$
What can we say about an hypersurface Einstein manifolds on $mathbb{R}^{n+1}$ when $ngeq 3$ ?
The manifolds is Einstein so Ricci tensor is a multiply of the metric $g$ on manifold:
$Ric=lambda g$ where $lambdain C^infty(M)$
If $S$ is the scalar curvature of $M$ then the relationship between $S$ and $lambda$ is
$lambda=frac{S}{n}$
Because the scalar curvature of $M$ is the trace of Ricci tensor.
So using the condition that $ngeq 3$ and using a contraction of the second Bianchi Identity we have that the scalar curvature of $M$ is costant so $lambda$ is costant.
Another condition that we have is that $M$ is an hypersurface of $mathbb{R}^{n+1}$ so we can use also the Codazzi-Mainardi Equation ti get some equation between the mean curvature $H$ of $M$, the metric $g$ and the second fondamental formula of $M$.
What can other we say about $M$, it is possible classify it?
general-topology geometry analysis riemannian-geometry riemann-surfaces
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
$endgroup$
$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34
add a comment |
$begingroup$
What can we say about an hypersurface Einstein manifolds on $mathbb{R}^{n+1}$ when $ngeq 3$ ?
The manifolds is Einstein so Ricci tensor is a multiply of the metric $g$ on manifold:
$Ric=lambda g$ where $lambdain C^infty(M)$
If $S$ is the scalar curvature of $M$ then the relationship between $S$ and $lambda$ is
$lambda=frac{S}{n}$
Because the scalar curvature of $M$ is the trace of Ricci tensor.
So using the condition that $ngeq 3$ and using a contraction of the second Bianchi Identity we have that the scalar curvature of $M$ is costant so $lambda$ is costant.
Another condition that we have is that $M$ is an hypersurface of $mathbb{R}^{n+1}$ so we can use also the Codazzi-Mainardi Equation ti get some equation between the mean curvature $H$ of $M$, the metric $g$ and the second fondamental formula of $M$.
What can other we say about $M$, it is possible classify it?
general-topology geometry analysis riemannian-geometry riemann-surfaces
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
$endgroup$
What can we say about an hypersurface Einstein manifolds on $mathbb{R}^{n+1}$ when $ngeq 3$ ?
The manifolds is Einstein so Ricci tensor is a multiply of the metric $g$ on manifold:
$Ric=lambda g$ where $lambdain C^infty(M)$
If $S$ is the scalar curvature of $M$ then the relationship between $S$ and $lambda$ is
$lambda=frac{S}{n}$
Because the scalar curvature of $M$ is the trace of Ricci tensor.
So using the condition that $ngeq 3$ and using a contraction of the second Bianchi Identity we have that the scalar curvature of $M$ is costant so $lambda$ is costant.
Another condition that we have is that $M$ is an hypersurface of $mathbb{R}^{n+1}$ so we can use also the Codazzi-Mainardi Equation ti get some equation between the mean curvature $H$ of $M$, the metric $g$ and the second fondamental formula of $M$.
What can other we say about $M$, it is possible classify it?
general-topology geometry analysis riemannian-geometry riemann-surfaces
general-topology geometry analysis riemannian-geometry riemann-surfaces
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
asked Jan 13 at 15:33
Federico FalluccaFederico Fallucca
1,959110
1,959110
$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34
add a comment |
$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34
$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34
add a comment |
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$begingroup$
Take a look here: researchgate.net/profile/Bang_Yen_Chen/publication/…
$endgroup$
– Moishe Cohen
Jan 13 at 18:06
$begingroup$
What I must search in this book? There are a lot of sections
$endgroup$
– Federico Fallucca
Jan 13 at 18:34