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**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.

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, PoincarĂ© conjecture |