Killing Reeb vector fields on some almost contact manifolds
Fortuné Massamba
Received January 17, 2025. Published online June 17, 2025.
Abstract: We discuss the geometric impact of a Killing vector field on certain almost contact manifolds with constant sectional curvatures. We prove that the additional structures given in A. De Nicola, G. Dileo, I. Yudin (2018) can be naturally constructed in the case of nearly Sasakian and nearly cosymplectic manifolds with constant sectional curvatures.
References: [1] V. N. Berestovskii, Y. G. Nikonorov: Killing vector fields of constant length on Riemannian manifolds. Sib. Math. J. 49 (2008), 395-407. DOI 10.1007/s11202-008-0039-3 | MR 2442533 | Zbl 1164.53346
[2] D. E. Blair: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203. Birkhäuser, Boston (2010). DOI 10.1007/978-0-8176-4959-3 | MR 2682326 | Zbl 1246.53001
[3] D. E. Blair, D. K. Showers, K. Yano: Nearly Sasakian structures. Kodai Math. Semin. Rep. 27 (1976), 175-180. DOI 10.2996/kmj/1138847173 | MR 0405287 | Zbl 0328.53036
[4] B. Cappelletti-Montano, A. De Nicola, G. Dileo, I. Yudin: Nearly Sasakian manifolds revisited. Complex Manifolds 6 (2019), 320-334. DOI 10.1515/coma-2019-0017 | MR 3987860 | Zbl 1427.53052
[5] B. Cappelletti-Montano, G. Dileo: Nearly Sasakian geometry and $SU(2)$-structures. Ann. Mat. Pura Appl. (4) 195 (2016), 897-922. DOI 10.1007/s10231-015-0496-9 | MR 3500312 | Zbl 1406.53028
[6] A. De Nicola, G. Dileo, I. Yudin: On nearly Sasakian and nearly cosymplectic manifolds. Ann. Mat. Pura Appl. (4) 197 (2018), 127-138. DOI 10.1007/s10231-017-0671-2 | MR 3747524 | Zbl 1384.53028
[7] S. Deshmukh, I. Al-Dayel: A note on nearly Sasakian and nearly cosymplictic structures of 5-dimensional spheres. Int. Electron. J. Geom. 11 (2018), 90-95. DOI 10.36890/iejg.545110 | MR 3875263 | Zbl 1409.53042
[8] S. Deshmukh, O. Belova: On killing vector fields on Riemannian manifolds. Mathematics 2021 (2021), Article ID 259, 17 pages. DOI 10.3390/math9030259
[9] D. Martelli, J. Sparks: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys. 262 (2006), 51-89. DOI 10.1007/s00220-005-1425-3 | MR 2200882 | Zbl 1112.53034
[10] F. Massamba, A. Nzunogera: A note on nearly Sasakian manifolds. Mathematics 2023 (2023), Article ID 2634, 20 pages. DOI 10.3390/math11122634
[11] Z. Olszak: Five-dimensional nearly Sasakian manifolds. Tensor, New Ser. 34 (1980), 273-276. MR 0597312 | Zbl 0448.53036
[12] B. O'Neill: Semi-Riemannian Geometry: With Applications to Relativity. Pure and Applied Mathematics 103. Academic Press, London (1983). MR 1567613 | Zbl 0531.53051
[13] X. Rong: Positive curvature, local and global symmetry, and fundamental groups. Am. J. Math. 121 (1999), 931-943. DOI 10.1353/ajm.1999.0036 | MR 1713297 | Zbl 0978.53073
[14] S. N. Shore: Astrophysical Hydrodynamics: An Introduction. Wiley, Weinheim (2007). DOI 10.1002/9783527619054
[15] J. A. Wolf: Spaces of Constant Curvature. AMS, Providence (2011). DOI 10.1090/chel/372 | MR 2742530 | Zbl 1216.53003
Affiliations: Fortuné Massamba, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa, e-mail: massfort@yahoo.fr, Massamba@ukzn.ac.za
