Quarter-Symmetric Nonmetric Connection on -Sasakian Manifolds

The object of the present paper is to study a quarter-symmetric nonmetric connection on a -Sasakian manifold. In this paper we consider the concircular curvature tensor and conformal curvature tensor on a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection. Next we consider second-order parallel tensor with respect to the quarter-symmetric non-metric connection. Finally we consider submanifolds of an almost paracontact manifold with respect to a quarter-symmetric non-metric connection. 1. Introduction In 1975, Golab [1] defined and studied quarter-symmetric connection in a differentiable manifold with affine connection. A linear connection on an -dimensional Riemannian manifold is called a quarter-symmetric connection [1] if its torsion tensor of the connection satisfies where is a 1 form and is a tensor field. In particular, if , then the quarter-symmetric connection reduces to a semisymmetric connection [2]. Thus the notion of quarter-symmetric connection generalizes the notion of the semisymmetric connection. If, moreover, a quarter-symmetric connection satisfies the condition for all , where is the Lie algebra of vector fields of the manifold , then is said to be a quarter-symmetric metric connection; otherwise it is said to be a quarter-symmetric nonmetric connection. After Golab [1], Rastogi [3, 4] continued the systematic study of quarter-symmetric metric connection. In 1980, Mishra and Pandey [5] studied quarter-symmetric metric connection in Riemannian, Kaehlerian, and Sasakian manifolds. In 1982, Yano and Imai [6] studied quarter-symmetric metric connection in Hermitian and Kaehlerian manifolds. In 1991, Mukhopadhyay et al. [7] studied quarter-symmetric metric connection on a Riemannian manifold with an almost complex structure . In 1997, Biswas and De [8] studied quarter-symmetric metric connection on a Sasakian manifold. In 2000, Ali and Nivas [9] studied quarter-symmetric connection on submanifolds of a manifold. Also in 2008, Sular et al. [10] studied quarter-symmetric metric connection in a Kenmotsu manifold. Let be a submanifold of an almost paracontact metric manifold with a positive definite metric . Let the induced metric on also be denoted by . The usual Gauss and Weingarten formulae are given, respectively, by where is the induced Riemannian connection on , is the second fundamental form of the immersion, and and are the tangential and normal parts of . From (1.4) and (1.5) one gets The submanifold of an almost paracontact manifold is called invariant (resp., anti-invariant) if for each point , (resp. . The