# Geodesic sphere in the octonion projective plane

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).

Yes; every geodesic sphere $$S_r$$, $$0, in the octonion (or Cayley) projective plane $$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 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).
• Thank you for confirming my question. How can we know that $Spin(9)$-homogeneous metrics on $S^{15}$ are canonical deformations of the round metric on $S^{15}$ with respect to the Hopf fibration? – Donghwi Seo Feb 4 '20 at 11:30
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.