Spectral identities and closed-geodesic periods

Nifa, Anass

doi:10.5281/zenodo.23587805

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Spectral identities and closed-geodesic periods

Anass NifaPublished: 1 July 2025Updated: 23 October 2026

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Abstract

We construct closed real-analytic locally symmetric nine-manifolds with equal scalar Laplace spectra and explicitly determine the infinite difference of their closed-geodesic period sets. Their common half-wave trace is real analytic near every period in this difference, although both geodesic flows are clean. In dimension eight, all finite-rank unitary flat connection spectra agree under a fixed $\mathbb Z/7\mathbb Z$ -marking, while the period sets differ. On a fixed closed simply connected eighteen-manifold, one of two real-analytic scalar-isospectral metrics has infinitely many primitive periods absent from the period set of the other. These spectral equalities follow from intrinsic equivariant identities for $\operatorname{U}(2n+1)$ and $\operatorname{Sp}(n)\times\operatorname{SO}(2n+2)$ , $n\ge1$ . For every central extension of a finite group with finitely generated kernel, prescribed finite families of central quotients occur as fundamental groups of closed homogeneous manifolds. Their canonical rough connection spectra agree for every finite-dimensional unitary representation of a common compact isotropy group. Simple connectivity varies in dimension eight, and rational formality in dimension seven. For a closed connected Riemannian manifold, marked finite-holonomy spectra determine the profinite conjugacy envelope of least lengths. Among compact connected Lie groups with bi-invariant metrics and a $\operatorname{U}(n)\times\operatorname{U}(1)$ marking, prescribed finite jets of the rate and logarithmic amplitude of the normalized equivariant heat trace, dimension and a volume lower bound characterize exactly these two models. The ambient weighted identity gives scalar-isospectral left-invariant normal homogeneous metrics on every $\operatorname{SU}(m)$ , $m\ge6$ , which are not locally isometric, answering Gordon–Sutton’s uniqueness question negatively.

MSC 2020

58J53Primary: Isospectrality
53C22Secondary: Geodesics in global differential geometry
53C30Secondary: Differential geometry of homogeneous manifolds
22E46Secondary: Semisimple Lie groups and their representations
55P62Secondary: Rational homotopy theory
20E18Secondary: Limits, profinite groups

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23587805
Subjects: Isospectral manifolds; closed-geodesic periods; wave traces; marked spectra; normal homogeneous spaces; Casimir operators; central extensions; canonical connections; profinite completion
Language: English

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Anass Nifa. Spectral identities and closed-geodesic periods. Preprint, Zenodo, v2 (2025). doi:10.5281/zenodo.23587805

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