Geodesic sphere in the octonion projective plane

Question

I am considering Laplacian eigenvalues of a geodesic sphere in the octonion projective plane and I would like to know the metric on a geodesic sphere.

Does the metric on a geodesic sphere in the octonion projective plane is a canonical variation associated with the Riemannian submersion $\pi : \mathbb{S}^{15} \rightarrow \mathbb{O}P^1 $ from the octonion Hopf fibration?

Loosely speaking, canonical variation on a metric associated with the Riemannian submersion with totally geodesic fibers is a variation on the given metric only in the fiber. More precise definition can be found in p. 191, Section 5, Berard-Bergery and Bourguignon - Laplacians and Riemannian submersions with totally geodesic fibres [Illinois Journ of Math, 1982] (MSN).

Answer 1

Yes; every geodesic sphere $S_r$, $0<r<\pi/2$, in the octonion (or Cayley) projective plane $Ca P^2$ is isometric to a canonical deformation of the round metric on $S^{15}$ with respect to the Hopf fibration $S^7\to S^{15}\to S^8(1/2)=Ca P^1$. Namely, decomposing the unit round metric on $S^{15}$ as $g_{hor}+g_{ver}$, that is, into horizontal and vertical parts in terms of this Riemannian submersion, we have that the metric on the geodesic sphere $S_r\subset CaP^2$ of radius $r$ is $$g_r=\sin^2 r\, (g_{hor}+\cos^2 r \, g_{ver}), \quad 0<r<\pi/2.$$ In particular, the Fubini-Study metric on $CaP^2$ can be written as a cohomogeneity one metric $g_{FS}=dr^2 + g_r$ along a radial unit speed geodesic $\gamma(r)$, with respect to the action of $Spin(9)$. This was probably first observed by Bourguignon and Karcher, at least in the complex and quaternion cases (see Section 6 here).

In particular, the Laplace eigenvalues of these geodesic spheres $(S_r,g_r)$ can be computed explicitly as a function of $r$, following the observations of Berard-Bergery and Bourguignon in the paper the OP cites. This is essentially carried out in Section 7 of this paper; one just needs to replace $t=\cos r$ and then also globally rescale with the factor $\sin^2 r$. One also should pay attention to the fact that not all combinations $(k,l)$ of horizontal and vertical eigenvalues appear, but this issue is addressed in a later paper of Besson and Bordoni (see here).

Answer 2

Here is what I remember, off the top of my head. Every geodesic on the octonionic projective plane (also called the octave projective plane) lies in a unique octonionic projective line, which is totally geodesic. The isometry group of the octonionic projective plane is a real form of $F_4$. Each octonionic projective line is acted on by a transitive subgroup of $F_4$, which is expressible as a group of linear fractional transformations (see John Baez on this), acting as the special orthogonal group $SO(9)$, so the metric is the standard one. You can see this by looking at the stabilizer of a point on the octonionic projective line, which you can see is $SO(8)$ by looking at the root system of $F_4$, I think.