I was reading about the Ricci Flow and the author used the following theorem: Let $0< w \leq \infty$, and let $g$ be a metric on the topological cylinder $(-w, w) \times S^n$ of the form $$g = \phi(z)^2 dz^2 + \psi(z)^2 g_{can}$$ where $\phi,\psi : (-w, w) \to \mathbb{R_+}$ and $g_{can}$ is the canonical round metric of radius $1$ on $S^n$. Then $g$ extends to a smooth metric on $S^{n+1}$ if and only if
$$ \int^w_{-w}\phi(r)dr < \infty$$ $$ \lim_{z \to \pm w} \psi(z) = 0 $$ $$\lim_{z \to \pm w} \frac{\psi'(z)}{\phi(z)} = \mp1 $$ and $$ \lim_{z \to \pm w} \frac{d^{2k}\psi(z)}{ds^{2k}} = 0 $$
for all $k \in \mathbb{N}$, where $ds$ is the element of arc length induced by $\phi$.
Could someone give me a reference where there is a proof of this result?
You can find this results and other facts on warped products over intervals in Peter Petersen's Riemannian Geometry- it lands in the first couple of chapters. I think a nice way to wrap up the final condition is that $\left( \frac{d\psi}{ds} \right)^2$ extends to a smooth function on $S^{n+1}$.