MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the so-called Berger spheres), i.e., for each $t>0$, let $S^{2n+1}_t$ (resp $S^{4n+3}_t$) be the Riemannian manifold $S^{2n+1}$ (resp $S^{4n+3}$) endowed with the metric obtained by scaling the round metric by a factor of $t^2$ in the direction of the fibers $S^1$ (resp $S^3$). For each $t>0$, they are the total space of a Riemannian submersion with totally geodesic fibers isometric to $tS^1$ (resp $tS^3$). In this context, the Laplacian $\Delta_t$ of the canonical variation can be related to the original Laplacian $\Delta_1$ in terms of the vertical Laplacian $\Delta_v$, by the formula $$\Delta_t=\Delta_1+(\tfrac{1}{t^2}-1)\Delta_v,$$ see Berard-Bergery and Bourguignon [Illinois Journ of Math, 1982]. The operator $\Delta_v$ acts on functions of the total space and is defined by $$(\Delta_v f)(p)=\Delta_{F_p}(f\vert_{F_p})$$ where $\Delta_{F_p}$ is the Laplacian of the fiber $F_p$.

Since the fibers are totally geodesic, the spectrum of $\Delta_v$ coincides with the spectrum of the Laplacian of the fiber, and all the above operators commute. In particular, $L^2$ of the total space admits a decomposition in simultaneous eigenspaces of $\Delta_t$ and $\Delta_v$. As a consequence, we have the following inclusion of spectra: $$Spec(\Delta_t) \subset Spec(\Delta_1)+(\tfrac{1}{t^2}-1) Spec(\Delta_F).$$

I am interested in computing $\lambda_1(t)$, the first non-zero eigenvalue of $\Delta_t$, for all $t$. By the above, for all $t>0$, we know $\lambda_1(t)$ is of the form $\mu_k+(\tfrac{1}{t^2}-1)\phi_j$ where $\mu_k\in Spec(\Delta_1)$ is an eigenvalue of the Laplacian on the original total space and $\phi_j\in Spec(\Delta_F)$ is an eigenvalue of the Laplacian of the fiber. The problem is that not all combinations of $\mu_k$'s and $\phi_j$'s give an eigenvalue of $\Delta_t$ and the first non-zero eigenvalue of $\Delta_t$ might be given by different combinations of $\mu_k$'s and $\phi_j$'s for each $t$. Recall that since total space and fibers are spheres, both $\mu_k$'s and $\phi_j$'s are easy to compute, namely the $k$th eigenvalue of the $m$-sphere is $k(k+m-1)$.

In the case of the first family with 1-dim fibers, this computation follows from a paper of Tanno [Tohoku Math Journ, 1979]. The trick is to look at a trajectory of the vector tangent to the $S^1$ fiber (a great circle) and solve the eigenvalue equation there (which becomes an ODE). Using this he proves that the only combinations of $\mu_k$'s and $\phi_j$'s permitted are when $0\leq j\leq k$ and $k-j$ is even. He then also finds explicit eigenfunctions when $k=j$; $j=1$ and $k$ odd; $j=2$ or $j=0$ and $k$ even. With these, it is very easy to compute $\lambda_1(t)$ for this family where the fibers are 1-dim.

However, I do not know any way of extending the result to the case of 3-dim fibers, or also the case of 7-dim fibers, $S^7\to S^{15}\to S^8$. It seems that Tanno's technique uses strongly the fact that the fibers are 1-dim. Any suggestion on how to compute this other first eigenvalues would be greatly appreciated.

share|cite|improve this question
Hi, Renato! It can be done probably this way. The Berger sphere $G/K=S^{2n+1}=SU(n+1)/SU(n)$ is a normal homogeneous space. Peter-Weyl decomposes $L^2(G)$ wrt left regular representation and this induces a decomposition of $L^2(G/K)$ into $G$-irreducible representations. The Laplacian acting on $L^2(G/K)$ is the same as the Casimir element of $\mathfrak g$ acting on those representations, and it acts as a scalar, for which there is an explicit and easy formula. The details are in Nolan Wallach's book on Harmonic Analysis (if I rememeber well), I think you can figure out. – Claudio Gorodski Jun 13 '11 at 1:34
In fact, the one-parameter family of metrics do not consist of normal homogeneous metrics, except for the basic one. It is still possible to adapt the method nontheless. – Claudio Gorodski Jun 13 '11 at 13:02
Hi Claudio, I believe you are refering to Lemma 5.6.4 on p. 124 of Wallach's book (the 1973 print), where it is shown that the Laplacian on a compact Lie group $G$ acts on the factors $\Gamma_\gamma E$ of the left regular representation decomposition as the scalar $c(\gamma)=\langle \Lambda_\gamma+\rho,\Lambda_\gamma+\rho\rangle-\langle\rho,\rho\rangle$, where $\gamma$ is the irreducible representation in question, $\Lambda_\gamma$ is the highest weight of $\gamma$ and $\rho$ is half the sum of the positive roots of $G$. However I still don't see how to apply this on any homogeneous space... – Renato G. Bettiol Jun 17 '11 at 16:32
If this is indeed the result you were mentioning, to obtain the Berger metrics on $SU(n+1)/SU(n)$ we consider a 1-parameter family of left-invariant metrics on $SU(n+1)$ (and the same should hold in the other homogeneous spaces $Sp(n+1)/Sp(n)$ and $Spin(9)/Spin(7)$). Even if we take, in the above formula for $c(\gamma)$, the inner product on $\mathfrak h^*$ (the Lie algebra of the maximal torus) to be the one induced by this 1-parameter family, why should $c(\gamma)$ be the eigenvalues of $\Delta_t$, even for the $\gamma$'s that have a non-zero vector fixed by $SU(n)$ (or $Sp(n),Spin(7)$)? – Renato G. Bettiol Jun 17 '11 at 16:39
No, the Laplacian of the normal homogeneous metric corresponds to the Casimir element and has the eigenavalues you described, but for the other metrics in the 1-parameter family you need to modify the Casimir element accordingly and correct the numerical value of the eigenvalues. Construct the Casimir element using a Killing orthonormal basis adapted to the submersion $SU(n+1)\to SU(n+1)/SU(n)$ and renormalize in one direction. For the normal homogeneous metric, Helgason's "Groups and geometric analysis" is also a good reference. – Claudio Gorodski Jun 17 '11 at 16:59
up vote 2 down vote accepted

Tanno, the same author of the paper above mentioned in the case of 1-dim fibers, has another paper one year later [Tanno, Shûkichi. Some metrics on a $(4r+3)$-sphere and spectra. Tsukuba J. Math. 4 (1980), no. 1, 99–105. ] in which he addresses exactly my original question, regarding the case of 3-dim fibers. As a consequence, one has an explicit formula for the first eigenvalue of the Laplacian $\Delta_t$ of $S^{4n+3}_t$ for all $t>0$. The technique is indeed related to Claudio's suggestion of using representation theory, since the Laplacian acts as the Casimir element. The case of $S^7\to S^{15}\to S^8$ can also be treated similarly, obtaining the first eigenvalue explicitly.

share|cite|improve this answer
Do you know if the "Berger" 7-sphere in $S^3\to S^7\to HP^1$ is the same as the physicists' "squashed 7-sphere" (discussed in §V.10 of Choquet-Bruhat et al. or these papers)? Comparing your spectrum inclusion with formula (13) of Nilsson-Pope, I would guess so...? – Francois Ziegler Sep 28 '15 at 1:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.