Problem: Consider the following warped product $M^{n+1}=\mathbb{R}\times_{f} \mathbb{P}^{n}$, where $\mathbb{P}$ is a complete n-dimensional Riemannian manifold, $f:\mathbb{R}\rightarrow\mathbb {R}_{+}$ is a smooth function positive(warping function),and the product manifold $\mathbb{R}\times \mathbb{P}^{n}$ is endowed with the complete Riemannian metric $$g=\pi^{*}_\mathbb{R}(dt^{2})+(f\circ\pi_\mathbb{R})^{2}\pi^{*}_\mathbb{P}(g_\mathbb{P})$$ Here $\mathbb\pi_\mathbb{R}$ and $\mathbb\pi_\mathbb{P}$ denote the projections onto the corresponding factor and ${g}_\mathbb{P}$ is the Riemannian metric on $\mathbb{P}^{n}$.Show that:
(a)The map $\tau:M\rightarrow \mathbb{J}\times \mathbb{P}$ given by $\tau(t,x)=(s(t),x)$ where $\mathbb{J=s(R)}$ and $$s(t)=s_{0}-\int_0^t\frac{1}{f(u)}du$$ is a reversing orientation isometry between $M$ and $\mathbb{J}\times \mathbb{P}$ endowed with the conformal metric $$g=\lambda^{2}(s)(ds^{2}+g_\mathbb{P})$$, where the conformal factor is $\lambda(s)=f(t(s))$. Suppose that $f(t)$ satisfies $$\int_0^{+\infty}\frac{1}{f}<{+\infty}$$ and $$\int_{-\infty}^0\frac{1}{f}={+\infty}$$ and take $s_{0}=\int_0^{+\infty}\frac{1}{f}$.Then, we have that $\mathbb{J=R_{+}}$ and therefore, $mathbb{P}$ acts sa a boundary at infinite of $M^{n+1}=\mathbb{R}\times_{f} \mathbb{P}^{n}$,as does $\mathbb {0}\times \mathbb{R}^{n}$ in $\mathbb{H}^{n+1}$, and the leaves $P_{t}=\mathbb{t}\times \mathbb{P}$ can be thought as horospheres in a fixed direction of $\mathbb{H}^{n+1}$. (b)Suppose that $\psi:\mathbb{\Sigma}\rightarrow\mathbb {R}\times_{f}\mathbb{P}$ is two-sided hypersurface with constant mean curvature $H$ and orientation $N$ (normal vector field). Show that the mean curvature of $\phi=\tau\circ\psi:\mathbb{\Sigma}\rightarrow\mathbb {J}\times_{f}\mathbb{P}$ is given by $$H^{*}=f(h)H+f´(h)\theta$$. Here $h=\pi_\mathbb{R}\circ\psi$ is the height function of hypersurface $\psi$ and $\theta=<N,\partial_{t}>$ is the angle function of $\psi$.
This problem is proposed in the article'' CONSTANT MEAN CURVATURE HYPERSURFACES IN WARPED PRODUCT SPACES'' due to Luis J. Alias and Marcos Dajczer.