SavedAdd your journal

Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 2

Weakly $\eta$-Einstein Almost Cosymplectic $3$-Manifolds

Ji-eun Lee(Chonnam National University, Department of Mathematics)0000-0003-0698-9596
Pages
—
Publication Date
2026-10-25
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

If the scalar curvature $r$ of a cosymplectic $3$-manifold $M$ is zero, then $M$ is Einstein, weakly Einstein, and flat.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We show that a strictly almost cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is weakly Einstein.
📊
Key Findings & Data:On the other hand, a cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is $\eta$-Einstein.
💡
Conclusion & Impact:If the scalar curvature $r$ of a cosymplectic $3$-manifold $M$ is zero, then $M$ is Einstein, weakly Einstein, and flat.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Almost cosymplectic manifold · weakly $\eta$-Einstein · weakly Einstein

Abstract

We show that a strictly almost cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is weakly Einstein. On the other hand, a cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is $\eta$-Einstein. If the scalar curvature $r$ of a cosymplectic $3$-manifold $M$ is zero, then $M$ is Einstein, weakly Einstein, and flat.

🔖

Cite This Article

Lee, J. (2026). Weakly $\eta$-Einstein Almost Cosymplectic $3$-Manifolds. International Electronic Journal of Geometry, Volume: 19 Issue: 2. https://doi.org/10.36890/iejg.1721313

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (21)

  1. Besse, A. L.: Einstein Manifolds, Springer.
  2. Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhäuser Boston, Inc., Boston, second edition, 2010.
  3. Cappelletti-Montano, B., Nicola, A. De., Yudin, I.: A survey on cosymplectic geometry, Rev. Math. Phys. 25 1343002 (2013).
  4. Chen, X.: On weakly Einstein almost contact manifolds, J. Korean Math. Soc. 57, 707-719 (2020).
  5. Cho, J. T.: Reeb flow symmetry on almost cosymplectic three-manifolds, Bull. Korean Math. Soc. 53 (4), 1249-1257 (2016).
  6. Cho, J. T., Chun, S. H., Euh, Y.: Weakly η-Einstein contact manifolds, Results in Mathematics, 77 (3), Paper No. 110, 16 pp (2022).
  7. Chun, S. H., Euh, Y.: Three-dimensional homogeneous contact metric manifold with weakly η-Einstein structures, Turk. J. Math., 47 (4), 1247- 1257 (2023).
  8. García-Río, E., Haji-Badali, A., Mariño-Villar, R., Vázquez-Abal, M. E.: Locally conformally flat weakly-Einstein manifolds, Arc. Math. 111, 549-559 (2018).
  9. Goldberg, S. I., Yano, K.: Integrability of almost cosymplectic structures, Pacific J. Math. 31, (2), 373-382 (1969).
  10. Inoguchi, J.: Characteristic Jacobi operator on almost cosymplectic 3-manifolds, Int. Electron. J. Geom. 12 (2), 276-299 (2019).
  11. Inoguchi, J., Lee, J.-E.: Almost cosymplectic 3-manifolds with pseudo-parallel characteristic Jacobi operator, Int. J. Geom. Methods Mod. Phys. 19 (8), Artcle ID 2250119 (2022).
  12. Inoguchi, J., Lee, J.-E.: Pseudo-symmetric almost cosymplectic 3-manifolds, Int. J. Geom. Methods Mod. Phys. 20 (10), Artcle ID 2350175 (2023).
  13. Lee, J.-E.: Weakly Einstein almost cosymplectic 3-manifolds, J. Korean Math. Soc. 62 (2), 421-442 (2025).
  14. Milnor, J.: Curvatures of left invariant metrics on Lie groups, Adv. Math. 21, 293-329 (1976).
  15. Olszak, Z.: On almost cosymplectic manifolds, Kodai Math. J. 4, 239-250 (1981).
  16. Öztürk, H., Aktan, N., Murathan, C.: Almost α-cosymplectic (κ, μ, ν)-spaces, arXiv:1007.0527v1 [math.DG]
  17. Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds, Differential Geom. Appl. 30, 49-58 (2012).
  18. Perrone, D.: Almost contact metric manifolds whose Reeb vector field is a harmonic section, Acta Math. Hungar. 138 (1-2), 102-126 (2013).
  19. Perrone, D.: Minimal Reeb vector fields on almost cosymplectic manifolds, Kodai Math. J. 36, 258-274 (2013).
  20. Besse, A. L.: Einstein Manifolds, Springer.
  21. Chen, X.: On weakly Einstein almost contact manifolds, J. Korean Math. Soc. 57, 707-719 (2020).