Published July 1, 2025 | Version v3
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Spectral identities and closed-geodesic periods

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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.

Mathematics Subject Classification (2020): Primary 58J53; Secondary 53C22, 53C30, 22E46, 55P62, 20E18.

First version: 1 July 2025. Updated 23 October 2026: version 2. Updated 9 November 2026: version 3 (151 pages).

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