Certain results on invariant submanifolds of an almost Kenmotsu (kappa ,mu ,nu )-space

Mehmet Atc̣Eken

doi:10.1007/s40065-021-00339-9

Abstract

In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has (kappa ,mu ,nu )-nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions kappa ,mu and nu behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu (kappa ,mu ,nu )-space to be totally geodesic under some conditions.

Highlights

It is well known that a (2n + 1)-dimensional contact metric manifold M admits an almost contact metric structure (φ, ξ, η, g), i.e., it admits a global vector field ξ, called the characteristic vector field or the Reeb vector field, its dual is η, a tensor φ of type (1, 1) and the Riemannian metric tensor g such that φ2 X = −X + η(X )ξ, η(ξ ) = 1, η ◦ φ = 0, (1)and g(φ X, φY ) = g(X, Y ) − η(X )η(Y ), (2)for all X, Y ∈ (T M), where (T M) denotes the set of differentiable vector fields on M [2]
The manifold M together with the structure tensor (φ, ξ, η, g) is called a contact metric manifold and we will denote it by M(2n+1)(φ, ξ, η, g) in the rest of this paper
2h X = ( ξ φ)X, for all X ∈ (T M), where ξ is the Lie-derivative in the direction of ξ

Summary

Introduction

J. Papantoniou introduced in [2] the notion of (κ, μ, ν)-contact metric manifold, its Riemannian curvature tensor R is given by We generalize the ambient space and investigate the conditions under which invariant pseudoparallel submanifolds of an almost Kenmotsu (κ, μ, ν)-space are totally geodesic. For all X, Y ∈ (T M) and V ∈ (T ⊥ M), where ∇ and ∇⊥ are the induced connections on M and (T ⊥ M) and σ and A are called the second fundamental form and shape operator of M, respectively, (T M) denotes the set differentiable vector fields on M.

Results

Conclusion