Spheres and projective spaces determined by the one-form spectrum

Nifa, Anass

doi:10.5281/zenodo.23984805

Preprint · v1

Spheres and projective spaces determined by the one-form spectrum

Anass Nifa5 November 2026

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Abstract

The ordinary one-form spectrum determines the standard spheres, real, complex and quaternionic projective spaces, and the Cayley plane among closed connected Riemannian manifolds. The same conclusion holds in the complementary degree without an orientability assumption; for the complex projective plane it holds in each degree $0 \leq p \leq 4$ . Above dimension four, partial spectral matchings suffice in degree one and in the complementary degree under quantitative bounds on omitted eigenvalue copies and matching cost. In sufficiently large dimension, a matching after independent finite omissions with error $o(\nu^{-1})$ , uniformly as the model eigenvalue $\nu \to \infty$ , determines every $d$ -dimensional model in a common set of form degrees. As $d \to \infty$ , the lower limit of their proportion among $0,\ldots,d$ is at least $\sqrt{8/21}$ . Heat identities and geodesic returns give averaged curvature bounds; for nonreal projective targets of dimension greater than four, equality in a periodic index estimate then determines the Jacobi operators.

MSC 2020

58J53Primary: Isospectrality
53C22Secondary: Geodesics in global differential geometry
53C26Secondary: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C55Secondary: Global differential geometry of Hermitian and Kählerian manifolds

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23984805
All versions: 10.5281/zenodo.23984804
Subjects: Hodge Laplacian; inverse spectral geometry; compact rank-one symmetric space; periodic geodesic flow
Language: English

Cite this paper

Anass Nifa. Spheres and projective spaces determined by the one-form spectrum. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.23984805

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References

As listed in the manuscript bibliography.

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DOI
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DOI
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DOI
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DOI · arXiv v2
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DOI · arXiv v1
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DOI
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K. Liu and H. Xu, A proof of the spectral rigidity for complex projective space, arXiv:2611.03536v1, 2 November 2026.
arXiv v1
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K. Mashimo, Spectra of the Laplacian on the Cayley projective plane, Tsukuba J. Math. 21 (1997), no. 2, 367–396. doi:10.21099/tkbjm/1496163248.
DOI
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A. Nifa, Spectral resolution and rigidity of spherical space forms, preprint, version 2, 23 October 2026. doi:10.5281/zenodo.23586002. Author’s version: CVGMT 8220.
DOI
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NIST Digital Library of Mathematical Functions, Release 1.2.8 of 2026-09-15, https://dlmf.nist.gov/.
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V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. (N.S.) 34 (1970), 269–285.
[40]
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DOI
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K. Shankar, R. Spatzier and B. Wilking, Spherical rank rigidity and Blaschke manifolds, Duke Math. J. 128 (2005), 65–81.
[43]
N. Sthanumoorthy, Spectral invariant of the zeta function of the Laplacian on Sp(𝑟 + 1)/Sp(1) × Sp(𝑟), Arch. Math. (Brno) 27 (1991), 95–104. DML-CZ 107408.
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C. J. Sutton, On the Poisson relation for compact Lie groups, Ann. Global Anal. Geom. 57 (2020), no. 4, 537–589. Preprint: arXiv:1606.07426.
arXiv
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S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. (2) 25 (1973), 391–403.
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S. Tanno, The spectrum of the Laplacian for 1-forms, Proc. Amer. Math. Soc. 45 (1974), 125–129.
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G. Tsagas, The spectrum of the Laplace operator for a special Riemannian manifold, Kodai Math. J. 4 (1981), 377–382.
[49]
C. Tsukamoto, Spectra of Laplace–Beltrami operators on SO(𝑛 + 2)/SO(2) × SO(𝑛) and Sp(𝑛 + 1)/Sp(1) × Sp(𝑛), Osaka J. Math. 18 (1981), 407–426.
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J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632.

[reference-2] [2]
M. Amann, C. Lange, and M. Radeschi, Odd-dimensional orbifolds with all geodesics closed are covered by manifolds, Math. Ann. 380 (2021), 1355–1386.

[reference-3] [3]
B. Ammann, R. Lauter, V. Nistor and A. Vasy, Complex powers and non-compact manifolds, Comm. Partial Differential Equations 29 (2004), no. 5–6, 671–705. doi:10.1081/PDE-120037329. Preprint: https://arxiv.org/abs/math/0211305v1.
DOI

[reference-4] [4]
V. I. Arnol’d, Characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967), no. 1, 1–13. doi:10.1007/BF01075861.
DOI

[reference-5] [5]
R. G. Bettiol, E. A. Lauret and P. Piccione, The first eigenvalue of a homogeneous CROSS, J. Geom. Anal. 32 (2022), article 76, 63 pp.

[reference-6] [6]
Q.-S. Chi, Curvature characterization and classification of rank-one symmetric spaces, Pacific J. Math. 150 (1991), no. 1, 31–42.

[reference-7] [7]
K. Cieliebak, N. Hingston and A. Oancea, Poincaré duality for loop spaces, Compos. Math. 161 (2025), 3140–3212. https://arxiv.org/abs/2008.13161v3.

[reference-8] [8]
R. L. Cohen, J. D. S. Jones and J. Yan, The loop homology algebra of spheres and projective spaces, in Categorical Decomposition Techniques in Algebraic Topology, Progr. Math. 215, Birkhäuser, 2003, 77–92.

[reference-9] [9]
R. L. Cohen, J. R. Klein and D. Sullivan, The homotopy invariance of the string topology loop product and string bracket, J. Topol. 1 (2008), 391–408.

[reference-10] [10]
Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979), 508–522.

[reference-11] [11]
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P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Publish or Perish, 1984.

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H. Gimperlein and G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ. 14 (2014), 49–83. doi:10.1007/s00028-013-0206-2. Preprint: arXiv:1302.6529v4.
DOI · arXiv v4

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A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), 157–198. doi:10.1007/BF02395060.
DOI

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A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.

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A. Hatcher, Vector Bundles & K-Theory, version 2.2, November 2017. https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf.

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DOI

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D. Jakobson, Y. Safarov and A. Strohmaier, The semiclassical theory of discontinuous systems and ray-splitting billiards, with an appendix by Y. Colin de Verdière, Amer. J. Math. 137 (2015), 859–906.

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arXiv v1

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DOI

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E. Kaneda, The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces II, J. Math. Kyoto Univ. 24 (1984), no. 1, 141–162. doi:10.1215/kjm/1250521390.
DOI

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R. Kuwabara, Spectrum of the Laplacian on vector bundles over C_{2π}-manifolds, J. Differential Geom. 27 (1988), no. 2, 241–258. doi:10.4310/jdg/1214441782.
DOI

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DOI

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L. Li and A. Strohmaier, The local counting function of operators of Dirac and Laplace type, J. Geom. Phys. 104 (2016), 204–228.

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P. Li, The spectral rigidity of complex projective spaces, revisited, Math. Z. 290 (2018), 1115–1143. doi:10.1007/s00209-018-2055-8. Preprint: arXiv:1608.02737v2.
DOI · arXiv v2

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P. Li, X. Sun, and A. Zhu, The spectral rigidity of Ricci soliton and Einstein-type manifolds, Comm. Anal. Geom., published online 22 October 2026, 32 pp. doi:10.4310/CAG.261018010944. Preprint: arXiv:2312.07259v1 (2023).
DOI · arXiv v1

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S. Lin and B. Schmidt, Real projective spaces with all geodesics closed, Geom. Funct. Anal. 27 (2017), 631–636, doi:10.1007/s00039-017-0407-x.
DOI

[reference-35] [35]
K. Liu and H. Xu, A proof of the spectral rigidity for complex projective space, arXiv:2611.03536v1, 2 November 2026.
arXiv v1

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K. Mashimo, Spectra of the Laplacian on the Cayley projective plane, Tsukuba J. Math. 21 (1997), no. 2, 367–396. doi:10.21099/tkbjm/1496163248.
DOI

[reference-37] [37]
A. Nifa, Spectral resolution and rigidity of spherical space forms, preprint, version 2, 23 October 2026. doi:10.5281/zenodo.23586002. Author’s version: CVGMT 8220.
DOI

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NIST Digital Library of Mathematical Functions, Release 1.2.8 of 2026-09-15, https://dlmf.nist.gov/.

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V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. (N.S.) 34 (1970), 269–285.

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D. Perrone, On the spectral rigidity of ℂℙⁿ, Proc. Amer. Math. Soc. 104 (1988), no. 3, 871–875. doi:10.1090/S0002-9939-1988-0964867-5.
DOI

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M. Radeschi and B. Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math. 210 (2017), 911–962.

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K. Shankar, R. Spatzier and B. Wilking, Spherical rank rigidity and Blaschke manifolds, Duke Math. J. 128 (2005), 65–81.

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N. Sthanumoorthy, Spectral invariant of the zeta function of the Laplacian on Sp(𝑟 + 1)/Sp(1) × Sp(𝑟), Arch. Math. (Brno) 27 (1991), 95–104. DML-CZ 107408.

[reference-44] [44]
C. J. Sutton, On the Poisson relation for compact Lie groups, Ann. Global Anal. Geom. 57 (2020), no. 4, 537–589. Preprint: arXiv:1606.07426.
arXiv

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S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. (2) 25 (1973), 391–403.

[reference-46] [46]
S. Tanno, The spectrum of the Laplacian for 1-forms, Proc. Amer. Math. Soc. 45 (1974), 125–129.

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G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Graduate Studies in Mathematics, vol. 99, Amer. Math. Soc., Providence, RI, 2009. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf.

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G. Tsagas, The spectrum of the Laplace operator for a special Riemannian manifold, Kodai Math. J. 4 (1981), 377–382.

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C. Tsukamoto, Spectra of Laplace–Beltrami operators on SO(𝑛 + 2)/SO(2) × SO(𝑛) and Sp(𝑛 + 1)/Sp(1) × Sp(𝑛), Osaka J. Math. 18 (1981), 407–426.

[reference-50] [50]
J.-Y. Wu, On almost isospectral manifolds, Indiana Univ. Math. J. 39 (1990), no. 4, 1373–1381.