In a previous question 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:

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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    $\begingroup$ Although Fabrice Baudoin's answer is very complete, you may also see Berger's spheres into a more general context (Riemannian submersions of the form $K/H\to G/H\to G/K$ for $H\subset K\subset G$ compact Lie groups) in Example 6.16 in Alexandrino-Bettiol's book "Lie groups and geometric aspects of isometric actions" and also Peter Petersen's book "Riemannian geometry" in Section 3 (the exact location depends on the edition). $\endgroup$ – emiliocba Sep 20 at 9:54

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.

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.

  • $\begingroup$ Thank you a lot, this was very helpful. Is there a reason for varying by $\frac{1}{\lambda^2}$ and not just $\lambda^2$? $\endgroup$ – horropie Sep 20 at 8:50
  • $\begingroup$ Just a matter of preference. It depends on authors, I think. $\endgroup$ – Fabrice Baudoin Sep 20 at 16:59

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