Help with identifying a reference for the time-invariant Kaluza-Klein metric

[user1139]

Homework Statement:: Please see below.
Relevant Equations:: Please see below.

I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric:

\begin{equation}
\mathrm{d}s^2_{(5)}=\lambda_{ab}\left(\mathrm{d}x^a+\omega^{a}_{\phantom{a}i}\mathrm{d}x^i\right)(\mathrm{d}x^b+\omega^{b}_{\phantom{b}j}\mathrm{d}x^j)+\frac{1}{\tau}h_{ij}\mathrm{d}x^i\mathrm{d}x^j.
\end{equation}

So far, I can only find sources for the Kaluza-Klein metric but not it's time-invariant version. Can someone point me to a relevant reference?

 

[MathematicalPhysicist]

I guess you tried the classical books on M-theory and Superstring of Joseph Polichinski, GWS green books or Kaku's books. (mind you, Kaku has also popular books but I am referring to his technical books).
Have you?

 

[user1139]

Yes @MathematicalPhysicist I have perused Polichinski's book in particular and could not find a mention of the time-invariant Kaluza-Klein metric.