It is known that the real hyperbolic space $\mathbb{H}^n$ has a warped product structure $g = dr^2 + \sinh^2 r\; ds^2_{n - 1} $, where $r \in (0, \infty)$. My question is, does the complex hyperbolic space $\mathbb{CH}^n$ also have a warped metric? Recall that a warped product metric on $(M, g) \times (N, h)$ is defined as $g + \varphi^2(x) h$, where $\varphi$ is a positive smooth function on $M$.
This is probably well-known, so this is mainly a reference request.
The answer is no. In polar coordinates (in an exponential chart) at any point of $\mathbb{CH}^n$, the complex hyperbolic metric reads $$ d r^2 + 4\sinh^2 (r) \theta^2 + 4 \sinh^2 (r/2) \gamma, $$ where $\theta$ is the standard contact form of the unit sphere $S^{2n-1}$, and $\gamma = d\theta(\cdot, i \cdot)$ is the associated Levi form, which is a positive definite metric on the contact distribution $H = \ker \theta$. Note that I have normalized the metric so that the sectional curvature lies between $-1$ (for complex lines) and $-1/4$ (for totally real plane). If the convention is that it lies between $-4$ and $-1$, then it reads $$ dr^2 + \sinh^2(2r)\theta^2 + \sinh^2(r) \gamma. $$
In fact, this form of the metric shows that geodesic spheres are not umbilical. They are like Berger's spheres: the metric is exploding in the direction of the Hopf fibration, which is also the direction of the Reeb vector field of $\theta$.