Differential Geometry of Manifolds

2023 №54(1)

Back to the list Download the article

On connections with torsion on nonholonomic para-Kenmotsu manifolds



The concept of a nonholonomic para-Kenmotsu manifold is intro­duced. A nonholonomic para-Kenmotsu manifold is a natural generaliza­tion of a para-Kenmotsu manifold; the distribution of a nonholonomic para-Kenmotsu manifold does not need to be involutive. Properly nonho­lonomic para-Kenmotsu manifolds are singled out, these are nonho­lono­mic para-Kenmotsu manifolds with non-involutive distribution. On an al­most (para-)contact metric manifold, we introduce a metric connec­tion with torsion, which is called a connection of Levi-Civita type in this pa­per. In the case of a nonholonomic para-Kenmotsu manifold, such a con­nection has a simpler structure than the Levi-Civita connection, and in so­me cases it turns out to be preferable from an applied point of view. A Le­vi-Civita type connection coincides with a Levi-Civita connec­tion if and only if a nonholonomic para-Kenmotsu manifold reduc­es to a para-Ken­motsu manifold. It is proved that a proper nonholonomic para-Ken­motsu manifold cannot carry the structure of an Einstein mani­fold with respect to a connection of the Levi-Civita type.


1. Bukusheva, A. V.: On the Schouten — Wagner tensor of a nonho­lonomic Kenmotsu manifold. Proceedings of the seminar on geometry and mathematical modeling, 5, 15—19 (2019).

2. Bukusheva, A. V.: Kenmotsu manifolds with a zero curvature dis­tribution. Tomsk State Univ. J. Math. Mech., 64, 5—14 (2020).

3. Bukusheva, A. V.: Geometry of nonholonomic Kenmotsu mani­folds. Izv. of Altai State Univ., 1 (117), 84—87 (2021).

4. Galaev, S. V.: Smooth distributions with admissible hypercomplex pseudo-Hermitian structure. Bulletin of Bashkir Univ., 21:3, 551—555 (2016).

5. Galaev, S. V.: Admissible hypercomplex structures on distributions of Sasakian manifolds. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. In­form., 16:3, 263—272 (2016).

6. Galaev, S. V.: Almost contact metric spaces with N-connection. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform, 15:3, 258—263 (2015).

7. Cappelletti-Montano, B., Erken, K. I., Murathan, C.: Nullity condi­tions in paracontact geometry. Differ. Geom. Appl., 30, 665—693 (2012).

8. Erken, K. I., Murathan, C.: A complete study of three-dimensional pa­racontact (k,μ,ν)-spaces. Int. J. Geom. Methods Mod. Phys., 14:7, 1750106 (2017).

9. Galaev, S. V.: Intrinsic geometry of almost contact Kahlerian mani­folds. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 31:1, 35—46 (2015).

10. Golab, S.: On semi-symmetric and quarter-symmetric linear con­nections. Tensor (N. S.), 29, 249—254 (1975).

11. Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J., 24, 93—103 (1972).

12. Olszak, Z.: The Schouten-van Kampen affine connection adapted to an almost (para)contact metric structure. Publ. Inst. Math. Nouv. ser., 94:108, 31—42 (2013).

13. Sato, I.: On a structure similar to the almost contact structure. Tensor (N. S.), 30, 219—224 (1976).

14. Sinha, B. B., Sai Prasad, K. L.: A class of almost paracontact met­ric manifolds. Bull. Cal. Math. Soc., 87, 307—312 (1995).

15. Wełyczko, J.: Slant curves in 3-dimensional normal almost par­acontact metric manifolds. Mediterr. J. Math., 11, 965—978 (2014).

16. Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Anal. Geom., 36, 37—60 (2009).