ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS

KRISHANU Mandal; U.C. De

Research Article

ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS

Year 2017, Volume: 5 Issue: 1, 11 - 23, 01.04.2017

KRISHANU Mandal U.C. De

Abstract

The aim of the present paper is to study 3-dimensional alpha-para Kenmotsu manifolds. First we consider 3-dimensional Ricci semisymmetric $\alpha$-para Kenmotsu manifolds and obtain some equivalent conditions. Next we study cyclic parallel Ricci tensor in 3-dimensional $\alpha$-para Kenmotsu manifolds. Moreover, we investigate $\eta$-parallel Ricci tensor in 3-dimensional $\alpha$-para Kenmotsu manifolds. Continuing our study, we consider locally $\phi$-symmetric 3-dimensional alpha-para Kenmotsu manifolds. Next, we study gradient Ricci solitons in 3-dimensional $\alpha$-para Kenmotsu manifolds. Finally, we give an example of a 3-dimensional $\alpha$-para Kenmotsu manifold which veries some results.

Keywords

$\alpha$-para Kenmotsu manifolds, Quasi constant curvature, Ricci semisymmetric, Cyclic parallel Ricci tensor, $\eta$-parallel Ricci tensor, Locally $\phi$-symmetric, Gradient Ricci solitons.

References

[1] T. Adati, Manifold of quasi-constant curvature II. quasi-umbilical hypersurfaces, TRU Math., 21(2)(1985), 221-226.
[2] T. Adati and T. Miyazawa, On a Riemannian space with recurrent conformal curvature, Tensor N.S., 18(1967), 348-354.
[3] T. Adati and Y. Wang, Manifolds of quasi-constant curvature I. A manifold of quasi-constant curvature and an S-manifold, TRU Math., 21(1)(1985), 95-103.
[4] C.L. Bejan and M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann. Glob. Anal. Geom., 46(2014), 117-127.
[5] A.M. Blaga, eta-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl., 20(2015), 1-13.
[6] Boeckx, E., Kowalski, O.and Vanhecke, L., Riemannian manifolds of conullity two, Singapore World Scientic Publishing, Singapore, 1996.
[7] C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Soc. 33(2010), 361-368.
[8] B.-Y. Chen and K. Yano, K., Hypersurfaces of a conformally at space, Tensor N.S., 26(1972), 318-322.
[9] B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical surveys and Monographs, Amer. Math. Soc. 110(2004).
[10] P. Dacko, On almost para-cosymplectic manifolds, Tsukuba J. Math., 28(2004), 193-213.
[11] U.C. De and S.K. Ghosh, Some properties of Riemannian spaces of quasi-constant curvature, Bull. Cal. Math. Soc., 93(2001), 27-32.
[12] A. Derdzinski, Compact Ricci solitons, Preprint.
[13] S. Erdem, On almost (para)contact (hyperbolic) metric manifolds and harmonicity of (eta; eta_0)- holomorphic maps between them, Houston J. Math., 28(2002), 21-45.
[14] A. Gray, Two classes of Riemannian manifolds, Geom. Dedicata 7(1978), 259-280.
[15] R.S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, Amer. Math. Soc.,(1988), 237-262.
[16] T. Ivey, Ricci solitons on compact 3-manifolds, Di. Geom. Appl. 3(1993), 301-307.
[17] U.-H. Ki and H. Nakagawa, A characterization of the cartan hypersurfaces in a sphere, Tohoku Math. J. 39(1987), 27-40.
[18] M. Kon, Invariant submanifolds in Sasakian manifolds, Math. Ann. 219(1976), 277-290.
[19] O. Kowalski, An explicit classication of 3-dimensional Riemannian spaces satisfying R(X; Y ) R = 0, Czech. Math. J., 46(1996), 427-474.
[20] M. Manev and M. Staikova, On almost paracontact Riemannian manifolds of type (n; n), J. Geom., 72(2001), 108-114.
[21] V.A. Mirzoyan, Structure theorems on Riemannian Ricci semisymmetic spaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat.,6(1992), 80-89.
[22] A.L. Mocanu, Les varietes a courbure quasi-constant de type Vranceanu, Lucr. Conf. Nat. de. Geom. Si Top., Tirgoviste, (1987).
[23] G. Nakova and S. Zamkovoy, Almost paracontact manifolds, (2009, reprint) arXiv:0806.3859v2.
[24] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint, http://arxiv.org/abs/math.DG/02111159.
[25] K. Srivastava and S.K. Srivastava, On a class of -Para Kenmotsu Manifolds, Mediterr. J. Math., 13 (2016), 391-399.
[26] Z.I. Szabo, Structure theorems on Riemannian spaces satisfying R(X; Y ) R = 0, the local version, J. Di. Geom., 17(1982), 531-582.
[27] T. Takahashi, Sasakian -symmetric spaces, Tohoku Math. J. 29(1977), 91-113.
[28] G. Vranceanu, Leconsdes Geometrie Dierential, Ed.de l'Academie, Bucharest, 4(1968).
[29] Y. Wang, On some properties of Riemannian spaces of quasi-constant curvature, Tensor N.S., 35(1981), 173-176.
[30] J. Welyczko, On Legendre curves in 3-dimensional normal almost paracontact metric manifolds, Result. Math., 54(2009), 377-387.
[31] J. Welyczko, Slant curves in 3-dimensional normal almost paracontact metric manifolds, Mediterr. J. Math., 11(2014), 965-978.
[32] A. Yildiz, U.C. De and M. Turan, On 3-dimensional f-Kenmotsu manifolds and Ricci solitons, Ukrainian Math. J. 65(2013), 684-693.
[33] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom., 36(2009), 37-60.

Year 2017, Volume: 5 Issue: 1, 11 - 23, 01.04.2017

KRISHANU Mandal U.C. De

Abstract

References

[1] T. Adati, Manifold of quasi-constant curvature II. quasi-umbilical hypersurfaces, TRU Math., 21(2)(1985), 221-226.
[2] T. Adati and T. Miyazawa, On a Riemannian space with recurrent conformal curvature, Tensor N.S., 18(1967), 348-354.
[3] T. Adati and Y. Wang, Manifolds of quasi-constant curvature I. A manifold of quasi-constant curvature and an S-manifold, TRU Math., 21(1)(1985), 95-103.
[4] C.L. Bejan and M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann. Glob. Anal. Geom., 46(2014), 117-127.
[5] A.M. Blaga, eta-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl., 20(2015), 1-13.
[6] Boeckx, E., Kowalski, O.and Vanhecke, L., Riemannian manifolds of conullity two, Singapore World Scientic Publishing, Singapore, 1996.
[7] C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Soc. 33(2010), 361-368.
[8] B.-Y. Chen and K. Yano, K., Hypersurfaces of a conformally at space, Tensor N.S., 26(1972), 318-322.
[9] B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical surveys and Monographs, Amer. Math. Soc. 110(2004).
[10] P. Dacko, On almost para-cosymplectic manifolds, Tsukuba J. Math., 28(2004), 193-213.
[11] U.C. De and S.K. Ghosh, Some properties of Riemannian spaces of quasi-constant curvature, Bull. Cal. Math. Soc., 93(2001), 27-32.
[12] A. Derdzinski, Compact Ricci solitons, Preprint.
[13] S. Erdem, On almost (para)contact (hyperbolic) metric manifolds and harmonicity of (eta; eta_0)- holomorphic maps between them, Houston J. Math., 28(2002), 21-45.
[14] A. Gray, Two classes of Riemannian manifolds, Geom. Dedicata 7(1978), 259-280.
[15] R.S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, Amer. Math. Soc.,(1988), 237-262.
[16] T. Ivey, Ricci solitons on compact 3-manifolds, Di. Geom. Appl. 3(1993), 301-307.
[17] U.-H. Ki and H. Nakagawa, A characterization of the cartan hypersurfaces in a sphere, Tohoku Math. J. 39(1987), 27-40.
[18] M. Kon, Invariant submanifolds in Sasakian manifolds, Math. Ann. 219(1976), 277-290.
[19] O. Kowalski, An explicit classication of 3-dimensional Riemannian spaces satisfying R(X; Y ) R = 0, Czech. Math. J., 46(1996), 427-474.
[20] M. Manev and M. Staikova, On almost paracontact Riemannian manifolds of type (n; n), J. Geom., 72(2001), 108-114.
[21] V.A. Mirzoyan, Structure theorems on Riemannian Ricci semisymmetic spaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat.,6(1992), 80-89.
[22] A.L. Mocanu, Les varietes a courbure quasi-constant de type Vranceanu, Lucr. Conf. Nat. de. Geom. Si Top., Tirgoviste, (1987).
[23] G. Nakova and S. Zamkovoy, Almost paracontact manifolds, (2009, reprint) arXiv:0806.3859v2.
[24] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint, http://arxiv.org/abs/math.DG/02111159.
[25] K. Srivastava and S.K. Srivastava, On a class of -Para Kenmotsu Manifolds, Mediterr. J. Math., 13 (2016), 391-399.
[26] Z.I. Szabo, Structure theorems on Riemannian spaces satisfying R(X; Y ) R = 0, the local version, J. Di. Geom., 17(1982), 531-582.
[27] T. Takahashi, Sasakian -symmetric spaces, Tohoku Math. J. 29(1977), 91-113.
[28] G. Vranceanu, Leconsdes Geometrie Dierential, Ed.de l'Academie, Bucharest, 4(1968).
[29] Y. Wang, On some properties of Riemannian spaces of quasi-constant curvature, Tensor N.S., 35(1981), 173-176.
[30] J. Welyczko, On Legendre curves in 3-dimensional normal almost paracontact metric manifolds, Result. Math., 54(2009), 377-387.
[31] J. Welyczko, Slant curves in 3-dimensional normal almost paracontact metric manifolds, Mediterr. J. Math., 11(2014), 965-978.
[32] A. Yildiz, U.C. De and M. Turan, On 3-dimensional f-Kenmotsu manifolds and Ricci solitons, Ukrainian Math. J. 65(2013), 684-693.
[33] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom., 36(2009), 37-60.

There are 33 citations in total.

Details

Subjects	Engineering
Journal Section	Articles
Authors	KRISHANU Mandal U.C. De
Publication Date	April 1, 2017
Submission Date	February 15, 2017
Acceptance Date	November 7, 2017
Published in Issue	Year 2017 Volume: 5 Issue: 1

Cite

APA	Mandal, K., & De, U. (2017). ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS. Konuralp Journal of Mathematics, 5(1), 11-23.
AMA	Mandal K, De U. ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS. Konuralp J. Math. April 2017;5(1):11-23.
Chicago	Mandal, KRISHANU, and U.C. De. “ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS”. Konuralp Journal of Mathematics 5, no. 1 (April 2017): 11-23.
EndNote	Mandal K, De U (April 1, 2017) ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS. Konuralp Journal of Mathematics 5 1 11–23.
IEEE	K. Mandal and U. De, “ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS”, Konuralp J. Math., vol. 5, no. 1, pp. 11–23, 2017.
ISNAD	Mandal, KRISHANU - De, U.C. “ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS”. Konuralp Journal of Mathematics 5/1 (April 2017), 11-23.
JAMA	Mandal K, De U. ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS. Konuralp J. Math. 2017;5:11–23.
MLA	Mandal, KRISHANU and U.C. De. “ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS”. Konuralp Journal of Mathematics, vol. 5, no. 1, 2017, pp. 11-23.
Vancouver	Mandal K, De U. ON 3-DIMENSIONAL $\alpha$-PARA KENMOTSU MANIFOLDS. Konuralp J. Math. 2017;5(1):11-23.

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