Spectral resolution and rigidity of spherical space forms

Let $n\ge5$ be odd. We prove quantitative rigidity from scalar Laplace spectra for two classes of round $n$-manifolds: homogeneous spherical space forms of even order and scalar lens spaces of odd order at least five. At fixed sufficiently small normalized frequency tolerance, an explicit strict bound on the lower limit of the least total unmatched multiplicity divided by $R^{(n+1)/2}$, along prescribed cutoffs $R$, forces a closed connected Riemannian manifold to be isometric to the reference space. Nonisometric lens spaces show that this exponent is sharp in general. A Besse homotopy $n$-sphere is round if it admits a constant-displacement isometry of finite odd order at least five.

Let $P$ be a self-adjoint classical elliptic operator of order one with positive scalar principal symbol, acting on a Hermitian bundle of positive rank over a closed connected $d$-manifold, $d\ge2$. If its spectrum concentrates modulo one near $s$ fixed phases, with vanishing tolerance and $o(R^d)$ exceptional eigenvalue copies along a sequence of cutoffs, then the time-$2\pi$ Hamiltonian map has order at most $s$, and there is a commuting quantizer $A$. Finer spectral resolutions give sharp upper bounds on the order of $P-A$. If $P-A$ is not smoothing, signed displacement profiles and Weyl mass determine the Liouville eigenvalue distribution of its normalized leading symbol.

For finite free isometric quotients of closed connected Zoll manifolds of dimension at least two, the Poisson relation is an equality, and the scalar spectrum determines the maximal dimension of returning components at each length. Among closed connected Riemannian $n$-manifolds whose distinct scalar eigenvalues have multiplicities $m_j=cj^{n-1}+O(j^{n-2})$, the possible positive coefficients are exactly $c=2^n/((n-1)!N)$, $N\in\mathbb Z_{>0}$. The largest value characterizes round real projective space up to scale.

Mathematics Subject Classification (2020): Primary 58J53; Secondary 53C22, 58E10, 55R20, 53D12.

First version: 2 September 2026. Updated 12 November 2026: version 3. 141 pages.