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.