In this paper, we characterized a new class of almost contact metric manifolds and established the equivalent conditions of the characterization identity in term of Kirichenko’s tensors. We demonstrated that the Kenmotsu manifold provides the mentioned class; i.e., the new class can be decomposed into a direct sum of the Kenmotsu and other classes. We proved that the manifold of dimension 3 coincided with the Kenmotsu manifold and provided an example of the new manifold of dimension 5, which is not the Kenmotsu manifold. Moreover, we established the Cartan’s structure equations, the components of Riemannian curvature tensor and the Ricci tensor of the class under consideration. Further,the conditions required for this to be an Einstein manifold have been determined.

This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {\Phi}-holomorphic sectional (G{\Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{\Phi}SH-curvature. In addition, the notion of {\Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {\eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.

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The present paper studies the generalized Φ −recurrent of Kenmotsu type manifolds. This is done to determine the components of the covariant derivative of the Riemannian curvature tensor. Moreover, the conditions which make Kenmotsu type manifolds to be locally symmetric or generalized Φ −recurrent have been established. It is also concluded that the locally symmetric of Kenmotsu type manifolds are generalized Φ −recurrent under suitable condition and vice versa. Furthermore, the study establishes the relationship between the Einstein manifolds and locally symmetric of Kenmotsu type manifolds.

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