Pointwise Slant Curves in Quasi-paraSasakian 3-Manifolds
- PDF / 563,895 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 77 Downloads / 211 Views
Pointwise Slant Curves in Quasi-paraSasakian 3-Manifolds K. Sood, K. Srivastava and S. K. Srivastava Abstract. This paper is devoted to the study of pointwise slant Frenet curves in a quasi-paraSasakian metric 3-manifold. We give a characterization result for the existence of such curves and determine their curvature and torsion in this class of manifold. Further, the characterizations for these curves having harmonic and C-parallel mean curvature vector field are derived. Mathematics Subject Classification. 53A55, 53B25, 53D15, 53C25, 53C50. Keywords. Pseudo-Riemannian metrics, paracontact structure, Frenet curves, pointwise slant curve, mean curvature vector field.
1. Introduction Paracontact pseudo-Riemannian manifolds were introduced in [1], as a natural (2m + 1)-dimensional counterpart to para-Hermitian manifolds, same as the contact Riemannian manifolds correspond to the Hermitian ones. Since the foundational paper [2], several authors, such as [3–9], studied paracontact pseudo-Riemannian manifolds, emphasizing analogies and differences with the contact Riemannian case. This study will focus on pointwise slant Frenet curves in quasi-paraSasakian 3-manifolds. These curves are natural generalization of slant Frenet curves. Slant curves in contact Riemannian geometry have been extensively studied by various researchers [10–13]. However, Welyczko [7] initiated the study 3 of slant curves in paracontact (pseudo-Riemannian) manifolds. Let M ; φ, η, ξ, g be an almost paracontact 3-manifold. Then a unit speed curve υ : I → M 3 is called a slant curve if g (υ (s) , ξ) = η (υ ) = ,
(1.1)
K. Sood: supported by DST, Ministry of Science and Technology, India through JRF [IF160490] DST/INSPIRE/03/2015/005481. K. Srivastava: supported by DST, Ministry of Science and Technology, India through WOS-A vide their File no. SR/WOS-A/PM20/2018.
0123456789().: V,-vol
114
Page 2 of 16
K. Sood et al.
MJOM
where is a constant function on open interval I and prime ( ) denotes differentiation with respect to arc length parameter s. The present paper is structured as follows. In Sect. 2, the basic definitions, subclasses of a normal almost paracontact metric 3-manifold N 3 and illustrations are given. Section 3 deals with pointwise slant Frenet curves (briefly, PS curves). The characterization result, expressions for curvature and torsion of such curves are obtained. Finally, in Sect. 4, PS curves having mean curvature vector field H, harmonic H and C-parallel H are characterized. Some examples are also constructed.
2. Normal Almost Paracontact (Pseudo-Riemannian) Manifolds 2.1. Paracontact Metric Manifolds A (2m + 1)-dimensional smooth manifold M2m+1 is called an almost paracontact manifold if M2m+1 is endowed with (φ, η, ξ)-structure satisfying: (i) η (ξ) = 1, φ2 = I − η ⊗ ξ, (ii) rank (φ) = 2m, η ◦ φ = 0, φξ = 0, (iii) φ induces an almost paracomplex structure on each fiber of horizontal distribution D = ker η, i.e., eigensubbundles D± corresponding to eigenvalues ±1 of φ have equal dimension m, where η is a paracontact f
Data Loading...
