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Spectral identities and closed-geodesic periods

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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 ℤ/7ℤ\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 U(2n+1)\operatorname{U}(2n+1) and Sp(n)×SO(2n+2)\operatorname{Sp}(n)\times\operatorname{SO}(2n+2), n≥1n\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 U(n)×U(1)\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 SU(m)\operatorname{SU}(m), m≥6m\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.24080426
All versions
10.5281/zenodo.23307577
Subjects
Isospectral manifolds; closed-geodesic periods; wave traces; marked spectra; normal homogeneous spaces; Casimir operators; central extensions; canonical connections; profinite completion
Language
English

Cite this paper

Anass Nifa. Spectral identities and closed-geodesic periods. Preprint, Zenodo, v3 (2025). doi:10.5281/zenodo.24080426

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