The spectrum of the Milnor-Gromoll-Meyer sphere. (Q1396367)

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. \textit{D. Gromoll} and \textit{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.