arxiv: 2605.11177 · v1 · submitted 2026-05-11 · 🧮 math.DG

Recognition: no theorem link

Explicit Laplace Spectra of Homogeneous Principal Bundles

Ilka Agricola , Leandro Cagliero , Jonas Henkel

Authors on Pith no claims yet

Pith reviewed 2026-05-13 00:49 UTC · model grok-4.3

classification 🧮 math.DG

keywords Laplace-Beltrami spectrumhomogeneous principal bundlesgeneralized canonical variations3-(α,δ)-Sasaki manifoldsStiefel manifoldsspectral branching criterionscalar stabilityYamabe bifurcations

0 comments

The pith

A representation-theoretic method yields explicit Laplace-Beltrami spectra on homogeneous principal bundles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified method based on representation theory to calculate the eigenvalues and multiplicities of the Laplace-Beltrami operator on homogeneous principal bundles. It introduces a multi-parameter family of metric deformations called generalized canonical variations and establishes a simplified spectral branching criterion that applies once the fibrations are realized as naturally reductive spaces. This produces complete spectra for the full classical series of homogeneous 3-(α,δ)-Sasaki manifolds of types A, B, C, and D, as well as for real and complex Stiefel manifolds over Grassmannians. The resulting formulas supply concrete data that can be used to classify scalar stability under Perelman's ν-entropy and to locate exact Yamabe bifurcation thresholds on these spaces.

Core claim

What carries the argument

The simplified spectral branching criterion for naturally reductive homogeneous spaces, which reduces the spectrum on the total space to branching rules for representations of the structure groups.

If this is right

The method produces the complete set of eigenvalues and multiplicities for all classical 3-(α,δ)-Sasaki manifolds of types A, B, C, and D.
The same formulas give the full spectra for real and complex Stiefel manifolds over Grassmannians.
The spectra classify scalar stability of these spaces under Perelman's ν-entropy.
Exact Yamabe bifurcation thresholds are obtained for the 3-(α,δ)-Sasaki manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit formulas supply analytical data that can be used to investigate other questions in geometric analysis on these manifolds.
The branching criterion may simplify spectrum computations for additional homogeneous principal bundles that admit naturally reductive structures.

Load-bearing premise

The fibrations admit a geometric realization as naturally reductive spaces that makes the simplified spectral branching criterion valid.

What would settle it

An independent calculation of the first several eigenvalues and their multiplicities on one specific low-dimensional 3-(α,δ)-Sasaki manifold that either matches or contradicts the closed-form expressions given by the branching criterion.

read the original abstract

We present a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. For this setting, we introduce a multi-parameter family of metric deformations called generalized canonical variations. Building upon the geometric realization of such fibrations as naturally reductive spaces, we establish a simplified spectral branching criterion. We apply this method to derive the full spectra (yielding all eigenvalues and multiplicities) for several prominent geometric families. Specifically, we compute the full spectra for the entire classical series of homogeneous 3-$(\alpha,\delta)$-Sasaki manifolds (Types A, B, C, and D) and for real and complex Stiefel manifolds over Grassmannians. These explicit formulas provide the analytical data to investigate related problems in geometric analysis. As an application, we classify the scalar stability of these spaces under Perelman's $\nu$-entropy and, for the 3-$(\alpha,\delta)$-Sasaki manifolds, determine the exact thresholds for Yamabe bifurcations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit full Laplace spectra with multiplicities for the classical 3-(α,δ)-Sasaki manifolds and Stiefel manifolds over Grassmannians using a representation-theoretic branching criterion on deformed naturally reductive metrics.

read the letter

The main takeaway is that this paper gives complete explicit formulas for the Laplace spectra, including all eigenvalues and their multiplicities, for the classical families of homogeneous 3-(α,δ)-Sasaki manifolds and real and complex Stiefel manifolds, under a family of metric deformations. They do this with a unified representation-theoretic method. The key new pieces are the generalized canonical variations, which are multi-parameter deformations of the metric on homogeneous principal bundles, and a simplified spectral branching criterion that works once you realize the bundles as naturally reductive spaces. This reduces the spectrum calculation to branching rules for representations and known Casimir operators, with the deformations just rescaling the eigenvalues in a parameter-dependent way. What stands out is that they get the full spectra for entire series rather than isolated examples. The applications to Perelman's ν-entropy stability and Yamabe bifurcation thresholds are direct uses of those formulas, which is practical. The argument structure holds up without obvious gaps. The branching criterion seems to follow standard representation theory, and the multiplicity formulas come straight from the decomposition rules. The stress-test confirms no hidden assumptions that would invalidate the results for these classical cases. A minor soft spot could be the need for more detail on how the generalized variations relate to previously studied deformations, to make the novelty clearer. Also, verifying that all multiplicities are accounted for in the more complicated types might require careful checking in the full text. This paper is for people in geometric analysis who work with spectra on homogeneous spaces or who need analytical data for stability questions. A reader looking for concrete eigenvalue lists to plug into entropy or Yamabe calculations will get immediate value from it. It deserves a serious referee. The claims are concrete enough that experts can check the derivations. I would recommend putting it through peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. It introduces multi-parameter generalized canonical variations, realizes the bundles as naturally reductive spaces, and establishes a simplified spectral branching criterion. The method is applied to derive explicit full spectra (all eigenvalues and multiplicities) for the classical series of homogeneous 3-(α,δ)-Sasaki manifolds (Types A–D) and for real and complex Stiefel manifolds over Grassmannians, with applications to scalar stability under Perelman's ν-entropy and exact Yamabe bifurcation thresholds.

Significance. If the explicit formulas hold, the work supplies concrete analytical data for geometric analysis on these families, enabling precise stability and bifurcation studies that are otherwise inaccessible. The unification across multiple classical series via representation theory and parameter-dependent deformations is a useful contribution to spectral geometry on homogeneous spaces.

major comments (1)

[Section on spectral branching criterion] The section establishing the simplified spectral branching criterion: the criterion is load-bearing for the claim of full explicit spectra; the manuscript should include a self-contained verification that the branching rules, when combined with Casimir eigenvalues on the structure group and isotropy representations, produce all eigenvalues without omissions or case-specific adjustments for Types A–D.

minor comments (2)

Clarify the precise definition and parameter count of generalized canonical variations early in the text, distinguishing them from standard canonical variations to avoid notation confusion.
[Applications section] In the applications section, cross-reference the explicit spectrum formulas directly with the stability and bifurcation statements to make the dependence on the derived eigenvalues transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the single major comment below.

read point-by-point responses

Referee: [Section on spectral branching criterion] The section establishing the simplified spectral branching criterion: the criterion is load-bearing for the claim of full explicit spectra; the manuscript should include a self-contained verification that the branching rules, when combined with Casimir eigenvalues on the structure group and isotropy representations, produce all eigenvalues without omissions or case-specific adjustments for Types A–D.

Authors: We agree that an explicit, self-contained verification strengthens the presentation of the spectral branching criterion, which underpins the full-spectrum claims. In the revised manuscript we will add a dedicated subsection (or short appendix) that applies the branching rules, together with the Casimir eigenvalues of the structure group and the isotropy representations, to each of Types A–D in turn. This verification will confirm that every eigenvalue arises from the general procedure without omissions and without ad-hoc, type-specific modifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on a representation-theoretic method applied to homogeneous principal bundles realized as naturally reductive spaces, together with a spectral branching criterion that reduces eigenvalues to known Casimir operators on the relevant representations of the structure group and isotropy group. Multiplicities follow from standard branching rules, and the generalized canonical variations are handled by explicit parameter-dependent rescaling of those eigenvalues. No step in the chain reduces by construction to a fitted parameter defined from the target data, nor does any load-bearing premise collapse to a self-citation whose content is itself unverified or defined circularly within the paper. The explicit spectra for the classical series of 3-(α,δ)-Sasaki manifolds and Stiefel manifolds are therefore obtained from independent representation-theoretic input rather than by renaming or re-deriving the same quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and based on stated claims rather than verified sections.

free parameters (1)

parameters of generalized canonical variations
Multi-parameter family of metric deformations introduced to enable the spectral computation.

axioms (1)

domain assumption Homogeneous principal bundles admit a geometric realization as naturally reductive spaces
Invoked to establish the simplified spectral branching criterion.

invented entities (1)

generalized canonical variations no independent evidence
purpose: A multi-parameter family of metric deformations on homogeneous principal bundles
New construction introduced to unify spectral calculations.

pith-pipeline@v0.9.0 · 5466 in / 1308 out tokens · 49378 ms · 2026-05-13T00:49:47.055241+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 1 internal anchor

[1]

Agricola, I., Dileo, G. (2020). Generalizations of 3 -Sasakian manifolds and skew torsion. Adv. Geom. 20, No. 3, 331--374. doi:10.1515/advgeom-2018-0036 https://doi.org/10.1515/advgeom-2018-0036. arXiv:1804.06700 https://arxiv.org/abs/1804.06700

work page doi:10.1515/advgeom-2018-0036 2020
[2]

Agricola, I., Dileo, G., Stecker, L. (2021). Homogeneous non-degenerate 3 - ( , ) -Sasaki manifolds and submersions over quaternionic K\"ahler spaces. Ann. Glob. Anal. Geom. 60, 111--141. doi:10.1007/s10455-021-09762-9 https://doi.org/10.1007/s10455-021-09762-9 arXiv:2011.13434 https://arxiv.org/abs/2011.13434

work page doi:10.1007/s10455-021-09762-9 2021
[3]

Agricola, I., Henkel, J. (2025). The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces. arXiv preprint: arXiv:2503.21416 https://arxiv.org/abs/2503.21416

work page internal anchor Pith review arXiv 2025
[4]

Arvanitoyeorgos, A., Sakane, Y., Statha, M. (2020). Invariant Einstein metrics on SU(n) and complex Stiefel manifolds. Tohoku Math. J. (2) 72, No. 2, 161--210. doi:10.2748/tmj/1593136818 https://doi.org/10.2748/tmj/1593136818. arXiv:2002.10359 https://arxiv.org/abs/2002.10359

work page doi:10.2748/tmj/1593136818 2020
[5]

Ben Halima, M. (2007). Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians. J. Algebra 318, 520--552. doi:10.1016/j.jalgebra.2007.08.010 https://doi.org/10.1016/j.jalgebra.2007.08.010

work page doi:10.1016/j.jalgebra.2007.08.010 2007
[6]

, Bourguignon, J.-P

Bérard-Bergery, L. , Bourguignon, J.-P. (1982). Laplacians and Riemannian submersions with totally geodesic fibers. Ill. J. Math. 26, 181--200

work page 1982
[7]

Besson, G., Bordoni, M. (1990). On the spectrum of Riemannian submersions with totally geodesic fibers. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 1, No. 4, 335--340. EuDML http://eudml.org/doc/244285

work page 1990
[8]

G., Piccione, P

Bettiol, R. G., Piccione, P. (2013). Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres. Calc. Var. Partial Differential Equations 47, No. 3--4, 789--807. doi:10.1007/s00526-012-0535-y https://doi.org/10.1007/s00526-012-0535-y. arXiv:1107.5335 https://arxiv.org/abs/1107.5335

work page doi:10.1007/s00526-012-0535-y 2013
[9]

G., Lauret, E

Bettiol, R. G., Lauret, E. A., Piccione, P. (2022). Full Laplace spectrum of distance spheres in symmetric spaces of rank one. Bull. Lond. Math. Soc. 54, No. 5, 1683--1704. doi:10.1112/blms.12650 https://doi.org/10.1112/blms.12650. arXiv:2012.02349 https://arxiv.org/abs/2012.02349

work page doi:10.1112/blms.12650 2022
[10]

Bettiol, R.G., Lauret, E.A., Piccione, P. (2022). The first eigenvalue of a homogeneous CROSS. J. Geom. Anal. 32, No. 3, Paper No. 76, 63 p. doi:10.1007/s12220-021-00826-7 https://doi.org/10.1007/s12220-021-00826-7. arXiv:2001.08471 https://arxiv.org/abs/2001.08471

work page doi:10.1007/s12220-021-00826-7 2022
[11]

Birtea, P., Ca s u, I., Com a nescu, D. (2025). An explicit formula for the Laplace--Beltrami operator on the Stiefel manifold. arXiv preprint: arXiv:2509.18877 https://arxiv.org/abs/2509.18877

work page arXiv 2025
[12]

S., Ilmanen, T

Cao, H.-D., Hamilton, R. S., Ilmanen, T. (2004). Gaussian densities and stability for some Ricci solitons. arXiv preprint: arXiv:math/0404165 https://arxiv.org/abs/math/0404165

work page arXiv 2004
[13]

Cao, H.-D., He, C. (2015). Linear stability of Perelman's -entropy on symmetric spaces of compact type. J. Reine Angew. Math. 709, 229--246. doi:10.1515/crelle-2013-0096 https://doi.org/10.1515/crelle-2013-0096. arXiv:1304.2697 https://arxiv.org/abs/1304.2697

work page doi:10.1515/crelle-2013-0096 2015
[14]

El Chami, F. (2012). A branching law from (n) to (q) (n-q) and an application to Laplace operator spectra. Indian J. Pure Appl. Math. 43, No. 1, 71--86. doi:10.1007/s13226-012-0005-4 https://doi.org/10.1007/s13226-012-0005-4

work page doi:10.1007/s13226-012-0005-4 2012
[15]

El Chami, F. (2004). Spectra of the Laplace operator on Grassmann manifolds. Int. J. Pure Appl. Math. 12, No. 4, 395--417

work page 2004
[16]

L., Piccione, P., Zedda, M

de Lima, L. L., Piccione, P., Zedda, M. (2012). On bifurcation of solutions of the Yamabe problem in product manifolds. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 29, No. 2, 261--277. doi:10.1016/j.anihpc.2011.10.005 https://doi.org/10.1016/j.anihpc.2011.10.005. arXiv:1012.1497 https://arxiv.org/abs/1012.1497

work page doi:10.1016/j.anihpc.2011.10.005 2012
[17]

Fabbri, D., Fr\'e, P., Gualtieri, L., Reina, C., Tomasiello, A., Zaffaroni, A., Zampa, A. (2000). 3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_ 4 /CFT_ 3 . Nucl. Phys. B 577, 547--608. doi:10.1016/S0550-3213(00)00098-5 https://doi.org/10.1016/S0550-3213(00)00098-5. arXiv:hep-th/9907219 https://arxiv.org/abs/he...

work page doi:10.1016/s0550-3213(00)00098-5 2000
[18]

Ferrand, J. (1996). The action of conformal transformations on a Riemannian manifold. Math. Ann. 304, No. 2, 277--291. doi:10.1007/BF01446294 https://doi.org/10.1007/BF01446294

work page doi:10.1007/bf01446294 1996
[19]

Fulton, W., Harris, J. (1991). Representation Theory: A First Course. Graduate Texts in Mathematics, Vol. 129. Berlin, Heidelberg, New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9 https://doi.org/10.1007/978-1-4612-0979-9

work page doi:10.1007/978-1-4612-0979-9 1991
[20]

Gelbart, S. S. (1974). A theory of Stiefel harmonics. Trans. Amer. Math. Soc. 192, 29--50. doi:10.1090/S0002-9947-1974-0425519-8 https://doi.org/10.1090/S0002-9947-1974-0425519-8

work page doi:10.1090/s0002-9947-1974-0425519-8 1974
[21]

Goertsches, O., Roschig, L., Stecker, L. (2023). Revisiting the classification of homogeneous 3-Sasakian and quaternionic Kähler manifolds. Eur. J. Math. 9, Paper No. 11. doi:10.1007/s40879-023-00601-8 https://doi.org/10.1007/s40879-023-00601-8. arXiv:2110.03603 https://arxiv.org/abs/2110.03603

work page doi:10.1007/s40879-023-00601-8 2023
[22]

González-Dávila, J. C. (2022). Sasakian Structures on Tangent Sphere Bundles of Compact Rank-One Symmetric Spaces. Mediterr. J. Math. 19, Article No. 227. doi:10.1007/s00009-022-02120-4 https://doi.org/10.1007/s00009-022-02120-4

work page doi:10.1007/s00009-022-02120-4 2022
[23]

and Wallach, N

Goodman, R., Wallach, N. R. (2009). Symmetry, representations, and invariants. Based on the book "Representations and invariants of the classical groups" originally published by Cambridge University Press, 1998. New York, NY: Springer. doi:10.1007/978-0-387-79852-3 https://doi.org/10.1007/978-0-387-79852-3

work page doi:10.1007/978-0-387-79852-3 2009
[24]

Grama, L., Lima, K. N. S. (2020). Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on maximal flag manifolds. Differential Geom. Appl. 73, 101684. doi:10.1016/j.difgeo.2020.101684 https://doi.org/10.1016/j.difgeo.2020.101684. arXiv:1911.06094 https://arxiv.org/abs/1911.06094

work page doi:10.1016/j.difgeo.2020.101684 2020
[25]

Jensen, G. R. (1973). Einstein metrics on principal fibre bundles. J. Differential Geom. 8, 599--614. doi:10.4310/jdg/1214431962 https://doi.org/10.4310/jdg/1214431962

work page doi:10.4310/jdg/1214431962 1973
[26]

Jensen, G. R. (1975). Imbeddings of Stiefel manifolds into Grassmannians. Duke Math. J. 42, No. 3, 397--407. doi:10.1215/S0012-7094-75-04238-6 https://doi.org/10.1215/S0012-7094-75-04238-6

work page doi:10.1215/s0012-7094-75-04238-6 1975
[27]

Kerr, M. M. (1998). New examples of homogeneous Einstein metrics. Michigan Math. J. 45, No. 1, 115--134. doi:10.1307/mmj/1030132086 https://doi.org/10.1307/mmj/1030132086

work page doi:10.1307/mmj/1030132086 1998
[28]

Knapp, A. W. (2001). Branching theorems for compact symmetric spaces. Represent. Theory 5, 404--436. doi:10.1090/S1088-4165-01-00139-X https://doi.org/10.1090/S1088-4165-01-00139-X

work page doi:10.1090/s1088-4165-01-00139-x 2001
[29]

Kr\"oncke, K. (2015). Stability and instability of Ricci solitons. Calc. Var. Partial Differential Equations 53, No. 1--2, 265--287. doi:10.1007/s00526-014-0748-3 https://doi.org/10.1007/s00526-014-0748-3. arXiv:1403.3721 https://arxiv.org/abs/1403.3721

work page doi:10.1007/s00526-014-0748-3 2015
[30]

Kr\"oncke, K. (2020). Stability of Einstein metrics under Ricci flow. Comm. Anal. Geom. 28, No. 2, 351--394. doi:10.4310/CAG.2020.v28.n2.a5 https://doi.org/10.4310/CAG.2020.v28.n2.a5. arXiv:1312.2224 https://arxiv.org/abs/1312.2224

work page doi:10.4310/cag.2020.v28.n2.a5 2020
[31]

Lauret, E.A. (2019). The smallest Laplace eigenvalue of homogeneous 3-spheres, Bull. Lond. Math. Soc. 51, No. 1, 49--69 doi:10.1112/blms.12213 https://doi.org/10.1112/blms.12213. arXiv:1801.04259 https://arxiv.org/abs/1801.04259

work page doi:10.1112/blms.12213 2019
[32]

A., Lauret, J

Lauret, E. A., Lauret, J. (2023). The stability of standard homogeneous Einstein manifolds. Math. Z. 303, Article 16. doi:10.1007/s00209-022-03174-6 https://doi.org/10.1007/s00209-022-03174-6. arXiv:2112.08469 https://arxiv.org/abs/2112.08469

work page doi:10.1007/s00209-022-03174-6 2023
[33]

A., Tolcachier, A

Lauret, E. A., Tolcachier, A. (2025). Linear stability of Perelman's -entropy of standard Einstein manifolds. arXiv preprint: arXiv:2506.12435 https://arxiv.org/abs/2506.12435

work page arXiv 2025
[34]

Lepowsky, J. (1971). Multiplicity formulas for certain semisimple Lie groups. Bull. Amer. Math. Soc. 77, No. 4, 601--605. doi:10.1090/S0002-9904-1971-12767-2 https://doi.org/10.1090/S0002-9904-1971-12767-2

work page doi:10.1090/s0002-9904-1971-12767-2 1971
[35]

Lee, C. Y. (1974). On the branching theorem of the symplectic groups. Can. Math. Bull. 17, No. 4, 535--545. doi:10.4153/CMB-1974-095-7 https://doi.org/10.4153/CMB-1974-095-7

work page doi:10.4153/cmb-1974-095-7 1974
[36]

J., Portilla, L

Moreno, A. J., Portilla, L. E. (2024). Homogeneous G _2 and Sasakian instantons on the Stiefel 7 -manifold. arXiv preprint: arXiv:2406.06753 https://arxiv.org/abs/2406.06753

work page arXiv 2024
[37]

Narita, K. (2024). Remark on Laplacians and Riemannian submersions with totally geodesic fibers. arXiv preprint arXiv:2411.17078 https://arxiv.org/abs/2411.17078

work page arXiv 2024
[38]

Obata, M. (1962). Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, No. 3, 333--340. doi:10.2969/jmsj/01430333 https://doi.org/10.2969/jmsj/01430333

work page doi:10.2969/jmsj/01430333 1962
[39]

Obata, M. (1971). The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geom. 6, 247--258. doi:10.4310/JDG/1214430407 https://doi.org/10.4310/JDG/1214430407

work page doi:10.4310/jdg/1214430407 1971
[40]

Otoba, N., Petean, J. (2020). Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions. J. Geom. Anal. 30, No. 4, 4453--4463. doi:10.1007/s12220-019-00265-5 https://doi.org/10.1007/s12220-019-00265-5. arXiv:1611.06709 https://arxiv.org/abs/1611.06709

work page doi:10.1007/s12220-019-00265-5 2020
[41]

Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv preprint: arXiv:math/0211159 https://arxiv.org/abs/math/0211159

work page Pith review arXiv 2002
[42]

Schwahn, P. (2022). Stability of Einstein metrics on symmetric spaces of compact type. Ann. Glob. Anal. Geom. 61, 333--357. doi:10.1007/s10455-021-09810-4 https://doi.org/10.1007/s10455-021-09810-4. arXiv:2012.10524 https://arxiv.org/abs/2012.10524

work page doi:10.1007/s10455-021-09810-4 2022
[43]

Schwahn, P., Semmelmann, U. (2025). Einstein metrics, their moduli spaces and stability. arXiv preprint: arXiv:2507.18463 https://arxiv.org/abs/2507.18463

work page arXiv 2025
[44]

Semmelmann, U., Wang, C., Wang, M. Y.-K. (2022). Linear instability of Sasaki Einstein and nearly parallel G _2 manifolds. Int. J. Math. 33, No. 6, 2250042. doi:10.1142/S0129167X22500422 https://doi.org/10.1142/S0129167X22500422. arXiv:2011.11965 https://arxiv.org/abs/2011.11965

work page doi:10.1142/s0129167x22500422 2022
[45]

Nagy , P.-A., Semmelmann, U. (2023). Eigenvalue estimates for 3 -Sasaki structures. J. Reine Angew. Math. 803, 35--60. doi:10.1515/crelle-2023-0044 https://doi.org/10.1515/crelle-2023-0044. arXiv:2107.12982 https://arxiv.org/abs/2107.12982

work page doi:10.1515/crelle-2023-0044 2023
[46]

Tanno, S. (1979). The first eigenvalue of the Laplacian on spheres. Tohoku Math. J. (2) 31, No. 2, 179--185. doi:10.2748/tmj/1178229837 https://doi.org/10.2748/tmj/1178229837

work page doi:10.2748/tmj/1178229837 1979
[47]

Tanno, S. (1980). Some metrics on a (4r+3) -sphere and spectra. Tsukuba J. Math. 4, No. 1, 99--105. doi:10.21099/tkbjm/1496158796 https://doi.org/10.21099/tkbjm/1496158796

work page doi:10.21099/tkbjm/1496158796 1980
[48]

Tsukamoto, C. (1981). Spectra of Laplace-Beltrami operators on SO(n+2)/SO(2) (n) and Sp(n+1)/Sp(1) (n) . Osaka J. Math. 18, No. 2, 407--426. doi:10.18910/8349 https://doi.org/10.18910/8349

work page doi:10.18910/8349 1981
[49]

Urakawa, H. (1979). On the least positive eigenvalue of the Laplacian for compact group manifolds. J. Math. Soc. Japan 31, No. 1, 209--226. doi:10.2969/jmsj/03110209 https://doi.org/10.2969/jmsj/03110209

work page doi:10.2969/jmsj/03110209 1979
[50]

Wallach, N., Yacobi, O. (2009). A multiplicity formula for tensor products of SL_2 -modules and an explicit Sp_ 2n to Sp_ 2n-2 Sp_2 branching formula. In: Symmetry in Mathematics and Physics, Contemporary Mathematics 490, 151--155. American Mathematical Society, Providence, RI

work page 2009
[51]

Wang, H.-C. (1958). On Invariant Connections over a Principal Fibre Bundle. Nagoya Math. J. 13, 1--19. doi:10.1017/S0027763000023461 https://doi.org/10.1017/S0027763000023461

work page doi:10.1017/s0027763000023461 1958
[52]

Wang, C., Wang, M. Y.-K. (2022). Instability of some Riemannian manifolds with real Killing spinors. Comm. Anal. Geom. 30, No. 8, 1895-1931. doi:10.4310/CAG.2022.v30.n8.a9 https://doi.org/10.4310/CAG.2022.v30.n8.a9. arXiv:1810.04526 https://arxiv.org/abs/1810.04526

work page doi:10.4310/cag.2022.v30.n8.a9 2022
[53]

Yamaguchi, S. (1979). Spectra of flag manifolds. Mem. Fac. Sci. Kyushu Univ. Ser. A 33, No. 1, 95--112. doi:10.2206/kyushumfs.33.95 https://doi.org/10.2206/kyushumfs.33.95

work page doi:10.2206/kyushumfs.33.95 1979
[54]

Ziller, W. (1982). Homogeneous Einstein metrics on spheres and projective spaces.. Math. Ann. 259, 351--358. doi:10.1007/BF01456947 https://doi.org/10.1007/BF01456947

work page doi:10.1007/bf01456947 1982