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The objective of this paper is to deal with Kenmotsu manifolds admitting $\eta$-Ricci-Yamabe solitons. First, it is proved that if a Kenmotsu manifold $M$ which admits an $\eta$-Ricci-Yamabe soliton, then the manifold $M$ is Einstein and is of constant scalar curvature. Then, some important characterizations, which classify Kenmotsu manifolds admitting such solitons, are obtained and an example given which supports our results.

The aim of this paper is to find some important classes of Einstein manifolds using conformal $\eta$-Ricci solitons and conformal $\eta$-Ricci almost solitons. We prove that a Kenmotsu metric as conformal $\eta$-Ricci soliton is Einstein if it is $\eta$-Einstein or the potential vector field $V$ is infinitesimal contact transformation or collinear with the Reeb vector field $\xi$. Next, we prove that a Kenmotsu metric as gradient conformal $\eta$-Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariants. Finally, we construct some examples to illustrate the existence of conformal $\eta$-Ricci soliton, gradient almost conformal $\eta$-Ricci soliton on Kenmotsu manifold.

The goal of this paper is to study conformal almost Ricci solitons within the framework of Kenmotsu manifolds. First, we demonstrate that if the potential vector field is Jacobi along the Reeb vector field, then the soliton reduces to a conformal Ricci soliton. If the manifold is $\eta$-Einstein Kenmotsu manifold, we show that either the manifold is of constant scalar curvature or the potential vector field is an infinitesimal contact transformation. In addition, if we consider the soliton vector field as a contact vector field, then either the gradient of $\lambda$ is pointwise collinear with the Reeb vector field or the manifold becomes $\eta$-Einstein. Lastly, we develop an example of a conformal almost Ricci soliton on the Kenmotsu manifold.

The goal of this paper is to study certain types of metric such as ∗-conformal Ricci–Yamabe soliton (RYS), whose potential vector field is torse-forming on Kenmotsu manifold. Here, we establish the conditions for solitons to be expanding, shrinking or steady and find the scalar curvature when the manifold admits ∗-conformal RYS on Kenmotsu manifold. Next, we developed the nature of the vector field when the manifold satisfies ∗-conformal RYS. Also, we have adorned some applications of torse-forming vector field in terms of ∗-conformal RYS on Kenmotsu manifold. We have also studied infinitesimal CL-transformation and Schouten–van Kampen connection on Kenmotsu manifold, whose metric is ∗-conformal RYS. We present an example of ∗-conformal RYS on three-dimensional Kenmotsu manifold, and verify some of our findings.

We prove that a Ricci almost soliton on a Kenmotsu manifold of dimension $>3$ reduces to an expanding Ricci soliton satifying certain condition on the potential vector field or on the soliton function. Next, we show that any Ricci almost soliton on a Kenmotsu manifold is trivial (Einstein) if the soliton vector leaves the contact form $\eta$ invariant. Finally, we classify (locally) a Kenmotsu manifold admitting an almost Yamabe soliton. Some examples have been constructed of almost Yamabe solitons on different class of warped product spaces.

This paper deals with the study of warped product pointwise bi-slant submanifolds of Kenmotsu manifolds with an example. The characterization for such submanifold is also discussed. An inequality of such submanifold is obtained and its equality case is also considered.

This paper examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci–Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $\eta$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes a Ricci–Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional ${(\kappa ,\mu )}^{\prime }$-AKM equipped with a gradient ARYS is either locally isometric to ${ℍ}^{n+1}(-4)\times {ℝ}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of three-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.

In the present work, Kenmotsu hypersurfaces in six-dimensional Hermitian submanifolds of Cayley algebra are considered. The Hermitian structure on such manifolds is induced by the means of the three-fold vector cross products in the octave algebra. A simple criterion of the minimality for a Kenmotsu hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra is established. It is also proved that the type number of a minimal Kenmotsu hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra is necessarily even.