Invariant Submanifolds of Kenmotsu Manifolds Admitting Quarter Symmetric Metric Connection

The object of this paper is to study invariant submanifolds M of Kenmotsu manifolds M̃ admitting a quarter symmetric metric connection and to show that M admits quarter symmetric metric connection. Further it is proved that the second fundamental forms σ and σ with respect to LeviCivita connection and quarter symmetric metric connection coincide. Also it is shown that if the second fundamental form σ is recurrent, 2-recurrent, generalized 2-recurrent, semiparallel, pseudoparallel, Ricci-generalized pseudoparallel and M has parallel third fundamental form with respect to quarter symmetric metric connection, then M is totally geodesic with respect to Levi-Civita connection. 1. Quarter symmetric metric connection The study of the geometry of invariant submanifolds of Kenmotsu manifolds is carried out by C.S. Bagewadi and V.S. Prasad [4], S. Sular and C. Ozgur [13] and M. Kobayashi [10]. The author [10] has shown that the submanifold M of a Kenmotsu manifold M̃ has parallel second fundamental form if and only if M is totally geodesic. The authors [4, 11, 13] have shown the equivalence of totally geodesicity of M with parallelism and semiparallelism of σ. Also they have shown that invariant submanifold of Kenmotsu manifold carries Kenmotsu structure and if K ≤ K̃, then M is totally geodesic. Further the author [13] have shown the equivalence of totally geodesicity of M , if σ is recurrent, M has parallel third fundamental form and σ is generalized 2-recurrent. Further the study has been carried out by B.S. Anitha and C.S. Bagewadi [2]. In this paper we extend the results to invariant submanifolds M of Kenmotsu manifolds admitting quarter symmetric metric connection. We know that a connection ∇ on a manifold M is called a metric connection if there is a Riemannian metric g on M if ∇g = 0 otherwise it is non-metric. In 1924, Friedman and J.A. Schouten [7] introduced the notion of a semi-symmetric linear 2000 Mathematics Subject Classification. 53D15, 53C21, 53C25, 53C40.