In the paper Multiply Twisted Products by Yong Wang, general definitions for so called *warped* and *twisted* products are given:

A **(singly) warped product** $B \times_b F$ of two pseudo-Riemannian manifolds $\left(B\,,g_B\right)$ and $\left(F\,,g_F\right)$ with a smooth function $b\,:\,B \to \left(0\,, \infty \right)$ is the product manifold $B \times F$ with the metric tensor $g = g_B \oplus b^2 g_F$. We call $\left(B\,,g_B\right)$ the **base** manifold, $\left(F\,,g_F\right)$ the **fiber** manifold and $b$ the warping function.

A **twisted product** $\left(M\,,g\right)$ is a product manifold $M = B \times_b F$, with a smooth function $b\,:\,B \times F \to \left(0\,, \infty \right)$ and the metric tensor $g = g_B \oplus b^2 g_F$.

The notion of *warped* products seems to be a rather standard definition, first introduced by O'Neill. I wonder if the definition of twisted products (and the generalizations of warped an twisted products) in the paper given above is also standard. Many theoretical physics papers sometimes talk about *warped* and *twisted* spacetimes in a rather handwavy way, so I am never really sure if they are talking about a mathematical definition or they just want to sound fancy.

For example, Compère talks in an overview paper about the Kerr/CFT correspondence about the following metric

\begin{align} \mathrm{d}s^2 = J\left(1+\cos^2\theta\right)\left[-r^2dt^2 + \frac{dr^2}{r^2} + d\theta^2 \right] + \frac{4 \sin^2\theta}{1+\cos^2\theta} \left(d\phi + rdt\right)^2\,, \end{align}

being a " warped and twisted product of $AdS_2 \times S^2$ " (J is just scaling the metric). This seems to be a different definition of a twisted product, as in Wangs paper twisted is a generalization of warped. Furthermore I can't neither make out the above metric to be $AdS_2 \times S^2$, nor that it is for $\theta = \frac{\pi}{2}$ a "twisted product of $AdS_2$ and a circle of constant radius" (as this paper suggests), as the suggested geometry should not have off-diagonal terms like $\mathrm{d}\phi \mathrm{d}t$ in the metric. For clarity, the metric of $AdS_2$ in Poincare coordinates is \begin{align} \label{eq:poincarepatch} ds^2 = - r^2dt^2 + \frac{dr^2}{r^2}\,, \end{align} where we set the curvature constant to $1$ to match the above expression.

My questions are:

- Am I missing coordinate transformations that make this obvious? As mentioned below this would have to make the metric diagonal.
- What is a standard definition for a twisted product of Pseudo-Riemannian manifolds?

EDIT: So as some comments say, the definition of a twisted product seems to be canonical. I need a coordinate transformation that gets rid of the off-diagonal term or at least a reference to a theorem that ensures the existence of such a coordinate transformation.

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