For $(N^n,g_N)$, define $(M^{n+1},g)=(\mathbb{R}\times N,\operatorname{d}r^2+\rho^2g_N)$. Write for convenience $\partial_0=\partial_r$, and $\overline{\bullet}, \bullet$ for quantities of $N,M$ respectively , then we have following results:

• Christoffel symbols:

  • ${\Gamma}^k_{ij}=\overline{\Gamma}^k_{ij}$
  • $\Gamma^0_{ij}=-\rho\rho'\overline{g}_{ij}$
  • $\Gamma_{0i}^j=\frac{\rho'}{\rho}\delta_{ij}$

  and all the other terms, i.e. with more than one 0-entry, are zero.

• Curvature tensors:

  • $R_{ijkl}=\rho^2\overline{R}{ijkl}+\rho^2(\rho')^2(\overline{g}{ik}\overline{g}{jl}-\overline{g}{il}\overline{g}_{jk})$
  • $R_{0ijk}=0$
  • $R_{0i0j}=\rho\rho''\overline{g}_{ij}$

  and all the other terms are zero.

• Ricci tensors:

  • $\operatorname{Ric}{00}=g^{ij}R{0ij0}=\frac{1}{\rho^2}\overline{g}^{ij}\cdot(-\rho\rho''\overline{g}_{ij})=-\frac{\rho''}{\rho}n$
  • $\operatorname{Ric}{ij}=g^{00}R{i00j+}g^{kl}R_{iklj}=-\rho\rho''\overline{g}{ij}+\overline{g}^{kl}\overline{R}{iklj}+(\rho')^2\overline{g}^{kl}(\overline{g}{ik}\overline{g}{jl}-\overline{g}{il}\overline{g}{jk})=\overline{\operatorname{Ric}}{ij}-\rho\rho''\overline{g}{ij}-(n-1)(\rho')^2\overline{g}_{ij}$

• scalar curvatures:

  • $S=g^{00}\operatorname{Ric}{00}+g^{ij}\operatorname{Ric}{ij}=\frac{1}{\rho^2}\overline{S}-2n\frac{\rho''}{\rho}-n(n-1)\frac{(\rho')^2}{\rho^2}$

As an example, consider $(N^{n-1},g_N)$ with $\operatorname{Ric}^N=\frac{n-2}{n-1}\lambda g_N,\lambda<0,$ if there is a function $\rho$ such that $\operatorname{Ric}=\lambda g$, then we get

$$ \left\{\begin{array}{ll}\lambda=-n\frac{\rho''}{\rho}\\\lambda\rho^2=\frac{n-2}{n-1}\lambda-\rho\rho''-(n-2)(\rho')^2\end{array}\right. $$

which gives $\rho=\cosh \left(\sqrt{\frac{\lambda}{1-n}}r\right)$.

For more results, see the notes by Petersen.

reference /source

https://link.springer.com/book/10.1007/978-3-319-26654-1