For exemplo i read $Ric_{\overline{M}}(N,N)$ = n in imersion $\phi: M^{n} \to \overline{M}=S^{n+1}$. Why? What's the relationship?
Ric = Ricci tensor e N is a normal vector field
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Where did you read this? Some further context would help. – Matheus Andrade Oct 29 '23 at 01:23
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For any vector in $S^{n+1}$, $\mathrm{Ric}(x,x) = n|x|^2$, but that is very specific here: $S^{n+1}$ has constant sectional curvature, and $\mathrm{Ric}(x,x)/|x|^2$ is the sum of the sectional curvatures of $n$ well-chosen planes containing $x$. – Didier Oct 29 '23 at 09:15
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(for any tangent vector of $S^{n+1}$, sorry) – Didier Oct 29 '23 at 09:50
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@Didier thanks, reference to "$Ric(x,x)=n ||x||^2"$ please! – Fabricio Gonçalves Oct 29 '23 at 14:44
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Context is Jacobi operator associated to second variational formula for area. consider a imersion $\phi: M \to S^{n+1}$ so operator associated to second variational formula for area is $J= -\Delta - |A|^2 - Ric_{\overline{M}}(N,N) \implies J= -\Delta - |A|^2 - n$ – Fabricio Gonçalves Oct 29 '23 at 14:55
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1@FabricioGonçalves any reference on the curvature tensor and the Ricci curvature of the unit sphere – Didier Oct 29 '23 at 19:37