I am trying to understand how warped product metrics work geometrically. For example, say one has a warped product space $(M \times \mathbb{R},g + \phi^2 \:dt^2)$ where the metric is in Cartesian coordinates and $\phi$ is the warping factor.
Now let's say we take a manifold given by the graph of a function $t = f(x,y,z)$ I am not sure what expression for the mean curvature of the graph would be, as I am not sure what the unit normal to the graph would be in the warped product metric.