## The spectrum of the Milnor-Gromoll-Meyer sphere.(English)Zbl 1040.58012

The author analyses the spectrum of the Milnor-Gromoll-Meyer sphere, i.e., a Riemannian manifold $$\Sigma^7$$ which is homeomorphic to the standard 7-sphere $$\mathbb S^7$$ but not diffeomorphic to $$\mathbb S^7$$. In particular, he shows that the eigenvalues $$0=\gamma_0<\gamma_1 \leq \gamma_2 \leq \dots$$ of $$\Sigma^7$$ are uniformly close to an explicitely given sequence $$\lambda_l$$, i.e., there exists a positive constant $$c$$ such that $$| \gamma_l - \lambda_l| \leq c$$ for all $$l=0,1,2, \dots$$. This result is not strong enough to “hear” the shape of $$\Sigma^7$$ but the author indicates that his calculations contribute to statistical properties of spectra.
The Milnor-Gromoll-Meyer sphere is defined as quotient $$\Sigma^7 = \Gamma \setminus Sp(2)$$ where $$Sp(n)$$ is the symplectic group for dimension $$n$$, i.e., the group of $$n \times n$$ quaternion matrices $$Q$$ such that $$QQ^*=Q^*Q=\text{Id}$$ with a normalized bi-invariant metric. Furthermore $$\Gamma$$ is the action of the quaternions on $$Sp(2)$$ given by $\Gamma(q,Q) = \begin{pmatrix} q & 0\\ 0 & q \end{pmatrix} Q \begin{pmatrix} \overline q & 0\\ 0 & 1 \end{pmatrix},$ [cf. D. Gromoll and W. Meyer, Ann. Math. (2) 100, 401–406 (1974; Zbl 0293.53015)].
More generally, the author calculates the spectrum of Riemannian manifolds $$M^7$$ given as quotient of $$Sp(2)$$ by certain actions of the quaternions. His examples comprise the exotic Milnor-Gromoll-Meyer $$7$$-sphere as well as three other $$7$$-spheres diffeomorphic to the standard sphere.
The main idea in the proof is to calculate the eigenspaces of the symplectic unitary group $$SpU(4) = Sp(4, \mathbb C) \cap U(4)$$, which is isomorphic to $$Sp(2)$$, and their subspaces of functions invariant under the actions.
Reviewer: Olaf Post (Aachen)

### MSC:

 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P20 Asymptotic distributions of eigenvalues in context of PDEs 43A75 Harmonic analysis on specific compact groups 57R60 Homotopy spheres, Poincaré conjecture

Zbl 0293.53015
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