[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5831/HMJ.2015.37.4.457

SOME CLASSES OF 3-DIMENSIONAL NORMAL ALMOST PARACONTACT METRIC MANIFOLDS

ERKEN, I. KUPELI (Department of Mathematics Uludag University Faculty of Science)

Publication Information

Honam Mathematical Journal / v.37, no.4, 2015 , pp. 457-468 More about this Journal

Abstract

The aim of present paper is to investigate 3-dimensional ${\xi}$ -projectively flt and $\tilde{\varphi}$ -projectively flt normal almost paracontact metric manifolds. As a first step, we proved that if the 3-dimensional normal almost paracontact metric manifold is ${\xi}$ -projectively flt then ${\Delta}{\beta}=0$ . If additionally ${\beta}$ is constant then the manifold is ${\beta}$ -para-Sasakian. Later, we proved that a 3-dimensional normal almost paracontact metric manifold is $\tilde{\varphi}$ -projectively flt if and only if it is an Einstein manifold for ${\alpha},{\beta}=const$ . Finally, we constructed an example to illustrate the results obtained in previous sections.

Keywords

normal almost paracontact metric manifold; curvarure tensor; ${\xi}$ -projectively flat; ${\varphi}$ -projectively flat; para-Sasakian manifold; Einstein manifold;

Citations & Related Records

Reference

1	D.V. Alekseevski, V. Cortes, A.S. Galaev, T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy, J. Reine Angew. Math. 635 (2009), 23-69.
2	D.V. Alekseevski, C. Medori, A. Tomassini, Maximally homogeneous para-CR manifolds, Ann. Glob. Anal. Geom. 30 (2006), 1-27. DOI
3	C. L. Bejan, M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann. Global Anal. Geom. 46(2) (2014), 117-127. DOI
4	D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics Vol. 203, Birkhauser, Boston, 2002.
5	B. Cappelletti-Montano, I. Kupeli Erken, C. Murathan, Nullity conditions in paracontact geometry, Diff. Geom. Appl. 30 (2012), 665-693. DOI
6	V. Cortes, C. Mayer, T. Mohaupt, F. Saueressing, Special geometry of Euclidean supersymmetry, 1. Vector multiplets. J. High Energy Phys. (2004) 03:028: 73.
7	V. Cortes, M.A. Lawn, L. Schafer, Affine hyperspheres associated to special para-Kahler manifolds, Int. J. Geom. Methods Mod. Phys. 3 (2006), 995-1009. DOI
8	P. Dacko, On almost para-cosymplectic manifolds, Tsukuba J. Math. 28 (2004), 193-213. DOI
9	U.C. De, A. K. Mondal, The structure of some classes of 3-dimensional normal almost contact metric manifolds, Bull. Malays. Math. Sci. Soc. 36 (2) (2013), 501-509.
10	S. Erdem, On almost (para)contact (hyperbolic) metric manifolds and harmonicity of ( $${\phi},{\varphi}$ )-holomorphic maps between them, Houston J. Math. 28 (2002), 21-45.
11	S. Kaneyuki , F. L.Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173-187. DOI
12	I. Kupeli Erken, C. Murathan, A Complete Study of Three-Dimensional Paracontact ( ${\kappa},{\mu},{\nu}$ )-spaces, Submitted. Available in Arxiv:1305.1511 [math. DG].
13	R. S. Mishra, Structures on a differentiable manifold and their applications, Chandrama Prakashan, Allahabad, 1984.
14	K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton Univ. Press, Princeton, N. J, 1953.
15	Z. Olszak, Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. XLVII (1986), 41-50.
16	J. Welyczko, On Legendre Curves in 3-Dimensional Normal Almost Paracontact Metric Manifolds, Result. Math. 54 (2009), 377-387. DOI
17	J. Welyczko, On basic curvature identities for almost (para)contact metric manifolds. Available in Arxiv: 1209.4731v1 [math. DG].
18	S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36 (2009), 37-60. DOI