# Literature Request: Berger Spheres and their Construction

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.

• 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). – emiliocba Sep 20 at 9:54

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 $$AdS_{2n+1} \to \mathbb{CH}_n$$.
• Thank you a lot, this was very helpful. Is there a reason for varying by $\frac{1}{\lambda^2}$ and not just $\lambda^2$? – horropie Sep 20 at 8:50