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Spectral resolution and rigidity of spherical space forms

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Abstract

Let n≥5n\ge5 be odd. We prove quantitative rigidity from scalar Laplace spectra for two classes of round nn-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)/2R^{(n+1)/2}, along prescribed cutoffs RR, 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 nn-sphere is round if it admits a constant-displacement isometry of finite odd order at least five.

Let PP 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 dd-manifold, d≥2d\ge2. If its spectrum concentrates modulo one near ss fixed phases, with vanishing tolerance and o(Rd)o(R^d) exceptional eigenvalue copies along a sequence of cutoffs, then the time-2π2\pi Hamiltonian map has order at most ss, and there is a commuting quantizer AA. Finer spectral resolutions give sharp upper bounds on the order of P−AP-A. If P−AP-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 nn-manifolds whose distinct scalar eigenvalues have multiplicities mj=cjn−1+O(jn−2)m_j=cj^{n-1}+O(j^{n-2}), the possible positive coefficients are exactly c=2n/((n−1)!N)c=2^n/((n-1)!N), N∈ℤ>0N\in\mathbb Z_{>0}. The largest value characterizes round real projective space up to scale.

MSC 2020

58J53Primary
Isospectrality
53C22Secondary
Geodesics in global differential geometry
58E10Secondary
Variational problems in applications to the theory of geodesics (problems in one independent variable)
55R20Secondary
Spectral sequences and homology of fiber spaces in algebraic topology
53D12Secondary
Lagrangian submanifolds; Maslov index

Mathematics Subject Classification 2020

Record

Version DOI
10.5281/zenodo.23586002
Subjects
Laplace spectrum; spherical space form; spectral rigidity; closed geodesics; Zoll manifold; Poisson relation
Language
English

Cite this paper

Anass Nifa. Spectral resolution and rigidity of spherical space forms. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23586002

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