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I am having trouble calculating the Ricci tensor. Suppose the product $I\times S^n$ has metric $$g=dt^2+f^2(t)g_{S^{n}}$$ where $g_{S^{n}}$ is the standard metric on $S^{n}$, what is the Ricci tensor of this metric? I have tried to do it for $n=2$ by writing the metric as $$g=\begin{pmatrix}1&0&0\\0&f^2&0\\0&0&f^2\sin^2\theta\end{pmatrix}$$ In a standard way, I calculated the Christoffel symbols and using the definition of Riemann tensor, of course, It is not wise to do similar thing in higher dimension. I am thinking that since the hypershpere is a manifold of constant sectional curvature 1 and the Ricci tensor is $$Ric=(n-1)g_{S^{n}}$$ Can we write the Ric on $I\times S^n$ from the Ric on $S^n$ and the product factor $f(t)$ quickly? But I don't know how to proceed. Any hint will be appreciated.

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