Narrow Results

In this paper, we study para-Sasakian manifold $(M,g)$ whose metric $g$ is an $\eta$-Ricci soliton $(g,V)$ and almost $\eta$-Ricci soliton. We prove that, if $g$ is an $\eta$-Ricci soliton, then either $M$ is Einstein and in such a case the soliton is expanding with $\lambda =2n$ or it is $D$-homothetically fixed $\eta$-Einstein manifold and in such a case the soliton is shrinking with $\lambda =-2n-4$. We show the same conclusion when the para-Sasakian manifold $(M,g)$ is of $\mathrm{\text{dim}}>3$ and $g$ is an almost $\eta$-Ricci soliton with $V$ as infinitesimal contact transformation. Finally, we prove that, if the para-Sasakian manifold $(M,g)$ of $\mathrm{\text{dim}}>3$ admits a gradient almost $\eta$-Ricci soliton with $V\ne 0$, then $M$ is Einstein. Suitable examples are constructed to justify our results.

Let M be an almost cosymplectic manifold such that the Reeb vector field is Killing. In this paper, it is proved that if the metric of M satisfying the $\eta$-Einstein condition is an $\eta$-Ricci soliton, then either M is Ricci flat or the potential vector field is an infinitesimal contact transformation. Also, a concrete example of cosymplectic $3$-manifold admitting non-trivial Ricci and $\eta$-Ricci solitons is constructed.

In the present paper, we study the invariant submanifolds of $f$-Kenmotsu manifolds. Firstly, we show that any invariant submanifold of $f$-Kenmotsu manifold is again $f$-Kenmotsu manifold and minimal. Then, we give some characterizations of totally geodesic submanifolds of $f$-Kenmotsu manifolds. We study 3-dimensional submanifold and prove that a 3-dimensional submanifold of $f$-Kenmotsu manifold is totally geodesic if and only if it is invariant. Also, $\eta$-Ricci soliton is considered on invariant submanifold of $f$-Kenmotsu manifolds. Lastly, the non-trivial examples are constructed to verify some of our obtained results.

In this paper, we study applications of some certain types of solitons such as conformal Ricci soliton, conformal $\eta$-Ricci–Yamabe soliton and $\eta$-Ricci soliton on Kählerian spacetime manifolds. Further, we have developed the characteristics of conformal Ricci soliton and conformal $\eta$-Ricci–Yamabe soliton on almost pseudo-symmetric Kählerian spacetime manifolds. Here, we have signalized the nature of solitons in terms of shrinking, steady or expanding and we have also presented the relationship between $\lambda$ and $\mu$ in terms of conformal $\eta$-Ricci–Yamabe soliton. Finally, we have embellished the classification of the potential function with respect to gradient $\eta$-Ricci soliton on Kählerian spacetime manifolds.

In this paper, we study ∗-Conformal $\eta$-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting ∗-Conformal $\eta$-Ricci soliton. We obtain some significant results on ∗-Conformal $\eta$-Ricci soliton in Sasakian manifolds satisfying $R(\xi ,X)\cdot S=0$, $S(\xi ,X)\cdot R=0$, $\overline{P}(\xi ,X)\cdot S$$=0$, where $\overline{P}$ is Pseudo-projective curvature tensor. The conditions for ∗-Conformal $\eta$-Ricci soliton on $\varphi$-conharmonically flat and $\varphi$-projectively flat Sasakian manifolds have been obtained in this paper. Lastly we give an example of five-dimensional Sasakian manifolds satisfying ∗-Conformal $\eta$-Ricci soliton.