On Generalized Weyl Conformal Curvature Tensor in Para-Kenmotsu  Manifolds: On Generalized Weyl Conformal Curvature Tensor

Mohd. Bilal; T. Raghuwanshi; M. K. Pandey; Abdul Haseeb; A. Goyal

Authors

Mohd. Bilal
T. Raghuwanshi
M. K. Pandey
Abdul Haseeb Jazan University
A. Goyal

Keywords:

Weyl conformal curvature tensor, para-Kenmotsu manifolds, Einstein manifold, generalized Weyl conformal curvature tensor

Abstract

In the present study, we consider a generalized Weyl conformal curvature tensor on para-Kenmotsu manifolds (briefly, $PK$-manifolds). First we describe certain vanishing properties of generalized Weyl conformal curvature tensor (brifly, GWC-curvature tensor) on a $PK$-manifold. Later, we study generalized Weyl conformally semi-symmetric $PK$-manifold that turns out to an Einstein manifold. Among others, it has been shown that the generalized Weyl conformally $\phi$-symmetric $PK$-manifold is of constant curvature or $dr (\psi)=0$.

References

T. Adati and T. Miyazava, On para-contact Riemannian manifolds, Tru Math. 13(2) (1977), 27–39.

A. Haseeb, R. Prasad, Certain results on Lorentzian para-Kenmotsu manifolds, Bol. Soc. Parana. Mat. 39(3) (2021),

–220.

M. Ali, A. Haseeb, F. Mofarreh, M. Vasiulla, Z-symmetric manifolds admitting Schouten tensor, Mathematics 10(22)

(2022), 1–10.

A. Haseeb, Some new results on para-Sasakian manifold with a quarter-symmetric metric connection, Facta

Universitatis (NIS), Ser. Math. Inform. 30(5) (2015), 765–776.

H. Weyl, Reine Infinitesimalgeometrie, Math. Z., 2(3–4) (1918), 384–411.

H. Weyl, Zur Infinitesimalgeometrie, Einordnung der projektiven und der konformen Auffassung, Gottingen

Nachrichten, (1921), 99–112.

A. M. Blaga, η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1–13.

Venkatesha, D. M Naik, H. A. Kumara, Conformal curvature tensor on para-contact metric manifolds, Matematicki

Vesnik 72(3) (2020), 215–225.

B. Cappelletti-Montano, I. Kupeli Erken, C. Murathan, Nullity conditions in paracontact geometry, Differ. Geom.

Appl. 30 (2012), 665–693.

I. Kupeli Erken, Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Periodica Math.

Hung. 80 (2020), 172–184.

I. Kupeli Erken and C. Murathan, A study of three-dimensional paracontact (k,μ,v)-spaces, Int. J. Geom. Methods

Mod. Phys. 14(7) (2017), 1750106.

C. A. Mantica, Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom.

Meth. Mod. Phys. 9(1) (2012), 1250004, 21 pages.

Z. Olszak, The Schouten-van Kampen affine connection adapted to an almost (para) contact metric structure, Publ.

Inst. Math. nouv. sér. 94(108) (2013), 31–42.

D. G. Prakasha, M. R. Amruthalakshmi, F. Mofarreh, A. Haseeb, Generalized Lorentzian Sasakian-space-forms with

M-projective curvature tensor, Mathematics 10(16) (2022), 1–14.

T. Raghuwanshi, S. K. Pandey, M. K. Pandey, A. Goyal, On generalized W2-curvature tensor of para-Kenmotsu mani-

folds, Filomat 36(3) (2022), 741–752.

S. S. Shukla, D. D. Singh, On conformal curvature tensor of (ε)-para Sasakian manifolds, SUT J. Math. 47(2) (2011),

–128.

M. Z. Petrovic, N. O. Vesic, M. Lj. Zlatanovic, Curvature properties of metric and semi-symmetric linear connections,

Quaestiones Mathematicae 45(10) (2022), 1603–1627.

T. Raghuwanshi, G. Pandey, M. K. Pandey, A. Goyal, On generalized pseudo-projective curvature tensor of para-

Kenmotsu manifolds, Bulletin of the Transilvania University of Barşov, Series III: Mathematics and Computer Science 1(63), no. 1 (2021), 143–160.

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

R. Prasad, L. Rathor, P. Gupta, A. Haseeb, A note on transversal hypersurfaces of para-Kenmotsu manifolds,

Proceedings of the Jangjeon Mathematical Society 26(1) (2023), 97–106.

A. Sardar, U. C. De, η-Ricci solitons on para-Kenmotsu manifolds, Differential Geometry Dynamical Systems 22 (2020),

–228.

Z. I. Szabo, Structure theorem on Riemannian space satisfying R(X,Y).R = 0. I. The local version, J. Diff.Geom. 17

(1982), 531–582.

T. Takahashi, Sasakian manifold with pseudo-Riemannian metric, Tohoku Math. J. 2(21) (1969), no. 2, 271–290.

K. Yano and M. Kon, Structures on manifolds, Series in Pure Math. Vol. 3 (1984), World Scientific.

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

On Generalized Weyl Conformal Curvature Tensor in Para-Kenmotsu Manifolds