We study a type of semisymmetric metric connection on a Riemannian manifold whose torsion tensor is almost pseudo symmetric and the associated 1-form of almost pseudo symmetric manifold is equal to the associated 1-form of the semisymmetric metric connection.

If the metric of an almost Kenmotsu manifold with conformal Reeb foliation is a gradient Ricci soliton, then it is an Einstein metric and the Ricci soliton is expanding. Moreover, let (M 2n+1 , φ, ξ, η, g) be an almost Kenmotsu manifold with ξ belonging to the (k, µ) ′-nullity distribution and h = 0. If the metric g of M 2n+1 is a gradient Ricci soliton, then M 2n+1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature −4 and a flat n-dimensional manifold, also, the Ricci soliton is expanding with λ = 4n.

The object of the present paper is to characterize Ricci semisymmetric almost Kenmotsu manifolds with its characteristic vector field ξ belonging to the (k, µ) -nullity distribution and (k, µ)-nullity distribution respectively. Finally, an illustrative example is given.

The present paper deals with invariant submanifolds of CR-integrable almost Kenmotsu manifolds. Among others it is proved that every invariant submanifold of a CR-integrable (k, µ)-almost Kenmotsu manifold with k < −1 is totally geodesic. Finally, we construct an example of an invariant submanifold of a CR-integrable (k, µ)-almost Kenmotsu manifold which is totally geodesic.

In this paper, we introduce a new type of curvature tensor named H-curvature tensor of type (1, 3) which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of H-curvature tensor. It is shown that a H-flat Lorentzian manifold is an almost product manifold. Then we study pseudo H-symmetric manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar curvature of the manifold vanishes and conversely. Moreover, we deal with pseudo H-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free H-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of (PHS)4.