In a [previous question](https://mathoverflow.net/questions/338565/explanation-for-squashing-and-stretching-lorentzian-analogue-of-berger-sphe) by me I asked about Berger spheres and their Lorentzian analogue, *squashed* $AdS\_3$ along Hopf fibres. It was well answered (by <https://mathoverflow.net/users/13268/ben-mckay>) but now I try to find literature about Berger spheres and their construction as in the answer, with very little success. For completeness I post how Berger spheres are constructed in the mentioned answer:



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> It is easier to simply write that the Hopf fibration is given by taking any connected 1-dimensional subgroup $K$ of $S^3$, and then the quotient $S^3 \to S^3/K$ is a Hopf fibration. Since the adjoint action of $S^3$ acts transitively on 1-dimensional subspaces of its Lie algebra (i.e. rotations of 3-dimensional Euclidean space act transitively on lines through the origin), $S^3$ acts transitively, by conjugation, on all connected 1-dimensional Lie subgroups. Hence the choice of $K$ is arbitrary. In your basis $z\_1$, $z\_2$,$z\_3$ for the Lie algebra of $S^3$, you could just take dual basis, say $\omega\_1$,$\omega\_2$,$\omega\_3$, and then your Hopf fibration can have fibers given by setting two linearly independent linear combinations of these two zero, say $0=\omega\_1=\omega\_2$, so the subgroup $K$ is then tangent to $z\_3$. We can make the Berger metric then be anything of the form $\omega\_1^2+\omega\_2^2+ \lambda \omega\_3^2$, carried around $S^3$ by left (or right, if you prefer) translation.
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Literature Request: Berger Spheres and their Construction
For a different viewpoint, the Berger Spheres or their Lorentzian analogues are well understood using the canonical variation of the metric associated with a Riemannian submersion with totally geodesic fibers, see Section 5 in [this paper](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256046790).


In the case of Berger Spheres, the picture is as follows. $\mathbf{U}(1)$ acts isometrically on $\mathbb S^{2n+1}$ The quotient space $\mathbb S^{2n+1} / \mathbf{U}(1)$ is the projective complex space $\mathbb{CP}^n$. The projection map $\pi : \mathbb S^{2n+1} \to \mathbb{CP}^n$ is a Riemannian submersion with totally geodesic fibers isometric to $\mathbf{U}(1)$. The sub-bundle $\mathcal{V}$ of $\mathbf{T}\mathbb S^{2n+1}$ formed by vectors tangent to the fibers of the submersion is referred to as the set of \emph{vertical directions}. The sub-bundle $\mathcal{H}$ of $\mathbf{T}\mathbb S^{2n+1}$ which is normal to $\mathcal{V}$ is referred to as the set of \emph{horizontal directions}. The standard metric $g$ of of $\mathbb S^{2n+1}$ can be split as
\begin{equation\*}
g=g\_\mathcal{H} \oplus g\_{\mathcal{V}},
\end{equation\*}
where the sum is orthogonal. We introduce the one-parameter family of Riemannian metrics:
\begin{equation}\label{eq-metric-B}
g\_{\lambda}=g\_\mathcal{H} \oplus \frac{1}{\lambda^2 }g\_{\mathcal{V}}, \quad \lambda >0,
\end{equation}
The Riemannian manifold $(\mathbb S^{2n+1}, g\_{\lambda})$ is the Berger sphere with parameter $\lambda >0$. The case $\lambda=1$ corresponds to the standard metric on $\mathbb S^{2n+1}$. 


**Fun fact**: When $\lambda \to 0$, in the Gromov-Hausdorff sense, $(\mathbb S^{2n+1}, g\_{\lambda})$ converges to $\mathbb S^{2n+1}$ endowed with the Carnot-Carath\'eodory metric (sub-Riemannian limit). When $\lambda \to \infty$, $(\mathbb S^{2n+1}, g\_{\lambda})$ converges to $\mathbb{CP}^n$ endowed with its standard Fubini-Study metric (adiabatic limit).


As is apparent, this construction generalizes to any Riemannian (or even semi-Riemannian) submersion. For instance the squashed AdS3 comes from the general anti-de Sitter semi Riemannian submersion $AdS\_{2n+1} \to \mathbb{CH}\_n$.


Besides the paper cited above, I remember that there is a part about the canonical variation of the metric associated with a Riemannian submersion in "Einstein manifolds" by Besse.


