A class of warped product submanifolds of Kenmotsu manifolds

Shyamal Kumar HUI,Mica S. STANKOVIC,Tanumoy PAL,Joydeb ROY

doi:10.3906/mat-1908-103

Abstract

In 2018 Naghi et al. studied warped product skew CR-submanifold of the form M = M1×f M⊥ of a Kenmotsu manifold M¯ throughout the paper , where M1 = MT ×Mθ and MT , M⊥, Mθ represents invariant, antiinvariant, proper slant submanifold of M¯ . Next, in 2019 Hui et al. studied another class of warped product skew CR-submanifold of the form M = M2 ×f MT of M¯ , where M2 = M⊥ × Mθ . The present paper deals with the study of a class of warped product submanifold of the form M = M3 ×f Mθ of M¯ , where M3 = MT × M⊥ and MT , M⊥, Mθ represents invariant, antiinvariant and proper pointwise slant submanifold of M¯ . A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product contact CR-submanifolds of the form M⊥ ×f MT , studied by Uddin et al. in 2017 and also generalizes the characterization of warped product semi-slant submanifolds of the form MT ×f Mθ , studied by Uddin in the same year. Beside that some inequalities on the squared norm of the second fundamental form are obtained which are also generalizations of the inequalities obtained in the just above two mentioned papers respectively.

Highlights

The third class of Tanno’s classification [30] is characterized by Kenmotsu [20]. This class is known as Kenmotsu manifold
Here we have considered the warped product submanifold of Mof the form M = M3 ×f Mθ, where M3 = MT × M⊥ and MT, M⊥, Mθ represents invariant, antiinvariant, and proper pointwise slant submanifolds of M, respectively
We study warped product submanifolds M = M3 ×f Mθ of Msuch that M3 = MT × M⊥ and ξ is tangent to M3, where MT, M⊥ and Mθ stands for invariant, antiinvariant, and proper pointwise-slant submanifolds of Mrespectively

Summary

Introduction

The third class of Tanno’s classification [30] is characterized by Kenmotsu [20]. This class is known as Kenmotsu manifold. Pointwise slant submanifolds in almost contact metric manifolds were studied in [24, 28]. A submanifold M of an almost contact metric manifold Mis said to be slant if for each nonzero vector X ∈ TpM , the angle θ between φX and TpM is constant, i.e. it does not depend on the choice of p ∈ M and X ∈ TpM. A submanifold M of an almost contact metric manifold Mis said to be pointwise slant [11] if for any nonzero vector X ∈ TpM at p ∈ M , such that X is not proportional to ξp , the angle θ(X) between φX and Tp∗M = TpM − {0} is independent of the choice of nonzero X ∈ Tp∗M. For any X ∈ Γ(T N1) and U ∈ Γ(T N2)

Submanifolds of M We consider a submanifold M of Msuch that

Warped product submanifolds of M

Generalized inequalities