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On almost cosymplectic generalized (k, μ)ʹ-spaces

  • Fortuné Massamba EMAIL logo
Published/Copyright: August 9, 2025
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Abstract

We introduce and study almost cosymplectic manifolds whose characteristic vector field ξ belongs to a generalized (k, μ)ʹ-nullity distribution, which involves the tensor = φ ◦ h. If ≠ 0, we prove that the smooth functions k and μ are uniquely determined by k < 0 and μ = ξ(ln λ), where λ=k. Moreover, we show that the spectrum of is {0, λ,−λ} and that the considered manifolds cannot be Ricci symmetric. Under certain conditions, we further prove that such manifolds are locally a warped product. Examples are also provided.


The author would like to thank the Institut des Hautes Études Scientifiques (IHES), Bures-Sur-Yvette (France) for its hospitality and support during the preparation of this paper. This work is partially supported by the National Research Foundation of South Africa.


  1. (Communicated by Tibor Macko)

References

[1] Besse, A. L.: Einstein Manifolds, Springer-Verlag, Berlin, 1987.10.1007/978-3-540-74311-8Search in Google Scholar

[2] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, Vol. 203, Birkhäuser Boston, Inc., Boston, MA., 2002.10.1007/978-1-4757-3604-5Search in Google Scholar

[3] Blair, D. E.—Koufogiorgos, T.—Papantoniou, B. J.: Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), 189–214.10.1007/BF02761646Search in Google Scholar

[4] Boeckx, E.: A full classication of contact (k, μ)-spaces, Illinois J. Math. 44 (2000), 212–219.10.1215/ijm/1255984960Search in Google Scholar

[5] Cappelletti-Montano, B.—Nicola, B. A. D.—Yudin, I.: A survey on cosymplectic geometry, Rev. Math. Phys. 25 (2013), Art. 1343002.10.1142/S0129055X13430022Search in Google Scholar

[6] Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publishing Co. Pte. Ltd., NJ, 2017.10.1142/10419Search in Google Scholar

[7] Carriazo, A.—Martín-Molina, V.: Almost cosymplectic and almost Kenmotsu (κ, μ, ν)-spaces, Mediterr. J. Math. 10(3) (2013), 1551–1571.10.1007/s00009-013-0246-4Search in Google Scholar

[8] Dacko, P.: On almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distribution, Balkan J. Geom. Appl. 5(2) (2000), 47–60.Search in Google Scholar

[9] Dacko, P.—Olszak, Z.: On conformally flat almost cosymplectic manifolds with Kählerian leaves, Rend. Sem. Mat. Univ. Politec. Torino 56(1) (1998), 89–103Search in Google Scholar

[10] Dacko, P.—Olszak, Z.: On almost cosymplectic (κ, μ, ν)-spaces. PDEs, submanifolds and affine differential geometry, 211–220, Banach Center Publ. 69, Polish Acad. Sci. Inst. Math., Warsaw, 2005.10.4064/bc69-0-17Search in Google Scholar

[11] Dileo, G.—Pastore, A. M.: Almost Kenmotsu manifolds with a condition of η-parallelism, Differential Geom. Appl. 27(5) (2009), 671–679.10.1016/j.difgeo.2009.03.007Search in Google Scholar

[12] Ghosh, A.—Sharma, R.—Cho, J. T.: Contact metric manifolds with η-parallel torsion tensor, Ann. Global Anal. Geom. 34(3) (2008), 287–299.10.1007/s10455-008-9112-1Search in Google Scholar

[13] Goldberg, S. I.—Yano, K.: Integrability of almost cosymplectic structures, Pacific J. Math. 31 (1969), 373–382.10.2140/pjm.1969.31.373Search in Google Scholar

[14] Kobayashi, S.—Nomizu, K.: Foundations of Differential Geometry, Vol. I, Wiley-Interscience Publishers, New York, 1963.Search in Google Scholar

[15] Koufogiorgos, T.—Markellos, M.—Papantoniou, V. J.: The harmonicity of the Reeb vector field on contact metric 3-manifolds, Pacific J. Math. 234(2) (2008), 325–344.10.2140/pjm.2008.234.325Search in Google Scholar

[16] Koufogiorgos, T.—Tsichlias, C.: On the existence of a new class of contact metric manifolds, Canad. Math. Bull. 43(4) (2000), 440–447.10.4153/CMB-2000-052-6Search in Google Scholar

[17] Maduna, S.—Massamba, F.: Certain class of almost cosymplectic manifolds with Kählerian leaves, Mediterr. J. Math. 20(3) (2023), Art. No. 163.10.1007/s00009-023-02343-zSearch in Google Scholar

[18] Pastore, A. M.—Saltarelli, V.: Generalized nullity distributions on almost Kenmotsu manifolds, Int. Electron. J. Geom. 4(2) (2011), 168–183.Search in Google Scholar

[19] Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds, Differential Geom. Appl. 30(1) (2012), 49–58.10.1016/j.difgeo.2011.10.003Search in Google Scholar

[20] Perrone, D.: Minimal Reeb vector fields on almost cosymplectic manifolds, Kodai Math. J. 36(2) (2013), 258–274.10.2996/kmj/1372337517Search in Google Scholar

[21] Olszak, Z.: Locally conformal almost cosymplectic manifolds, Colloq. Math. 57 (1989), 73–87.10.4064/cm-57-1-73-87Search in Google Scholar

[22] Olszak, Z.: On almost cosymplectic manifolds, Kodai Math. J. 1 (1981), 239–250.10.2996/kmj/1138036371Search in Google Scholar

[23] Olszak, Z.: Almost cosymplectic manifolds with Kählerian leaves, Tensor New Ser. 46 (1987), 117–124.Search in Google Scholar

[24] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, Pure Appl. Math., Vol. 103, Academic Press, New-York, 1983.Search in Google Scholar

[25] ÖZtüRk, H.—Aktan, N.—Murathan, C.: Almost α-Cosymplectic (κ, μ, ν)-Spaces, https://arxiv.org/abs/1007.0527.Search in Google Scholar

[26] Szabo, Z. I.: Structure theorems on Riemannian spaces satisfying R(X, Y) · R = 0 I: The local version, J. Differential Geometry 17(4) (1982), 531–582.10.4310/jdg/1214437486Search in Google Scholar

[27] Szabo, Z. I.: Structure theorems on Riemannian spaces satisfying R(X, Y) · R = 0 II: The global version, Geom. Dedicata 19 (1985), 65–10810.1007/BF00233102Search in Google Scholar

[28] Yildiz, A.—De, U. C.: A classification of (κ, μ)-contact metric manifolds, Commun. Korean Math. Soc. 27(2) (2012), 327–339.10.4134/CKMS.2012.27.2.327Search in Google Scholar

[29] Wang, Y.: Almost co-Kähler manifolds satisfying some symmetry conditions, Turkish J. Math. 40(4) (2016), 740–752.10.3906/mat-1504-73Search in Google Scholar

[30] Wang, Y.: Locally symmetric almost co-Kähler 5-manifolds with Kählerian leaves, Bull. Korean Math. Soc. 55(3) (2018), 789–798.Search in Google Scholar

[31] Wang, W.: Almost cosymplectic (k, μ)-metrics as η-Ricci solitons, J. Nonlinear Math. Phys. 29(1) (2022), 58–72.10.1007/s44198-021-00019-4Search in Google Scholar

Received: 2024-09-24
Accepted: 2025-04-01
Published Online: 2025-08-09
Published in Print: 2025-08-26

© 2025 Mathematical Institute Slovak Academy of Sciences

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