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).

  • $\begingroup$ What does "Does the metric on a geodesic sphere in the octonion projective plane is a canonical variation" mean? Does it mean "Is the metric on a geodesic sphere in the octonion projective plane a canonical variation"? $\endgroup$
    – John Baez
    Mar 9, 2021 at 19:09

2 Answers 2


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).

  • $\begingroup$ 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? $\endgroup$ Feb 4, 2020 at 11:30
  • $\begingroup$ @DonghwiSeo: This follows from analyzing the isotropy representation of the action. Namely, the subgroup of Spin(9) that fixes a given point on S^15 is isomorphic to Spin(7), and its isotropy representation splits as a direct sum of 2 irreducible modules, of dimensions 8 and 7. With a little work, you can see that these are respectively the horizontal and vertical space of the Hopf fibration. Sorry for the very late reply, NYC has been hard to live in lately.. $\endgroup$ Apr 30, 2020 at 0:56

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.


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