The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field $\xi$ in connection with conformal Ricci–Yamabe metric and conformal $\eta$-Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or shrinking. We also discuss conformal Ricci–Yamabe soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid and radiation era. Furthermore, we design conformal $\eta$-Ricci–Yamabe soliton to find its characteristics in a perfect fluid spacetime and lastly acquired Laplace equation from conformal $\eta$-Ricci–Yamabe soliton equation when the potential vector field $\xi$ of the soliton is of gradient type. Overall, the main novelty of the paper is to study the geometrical phenomena and characteristics of our newly introduced conformal Ricci–Yamabe and conformal $\eta$-Ricci–Yamabe solitons to apply their existence in a perfect fluid spacetime.

The outline of this research article is to initiate the development of a ∗-conformal $\eta$-Ricci–Yamabe soliton in $\alpha$-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of $\alpha$-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal $\eta$-Ricci–Yamabe soliton equation when the potential vector field $\xi$ of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional $\alpha$-cosymplectic metric as a ∗-conformal $\eta$-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results.

Giving the definitions of one-parameter family of frontals in lightcone and one-parameter family of spacelike Legendrian curves in ${\Delta }_2$ and ${\Delta }_3$, and further using the variability condition and the tangency condition, the definitions of envelopes of these geometric objects are presented. The aim of this work is to explore the criterion conditions on the envelopes of a one-parameter family of frontals related to ${\Delta }_2$ and ${\Delta }_3$ in three pseudo-spheres. Thereby the characterizations of these envelopes are described via Envelope Theorems. The geometric relations among these envelopes are discussed in detail. It is demonstrated that the ${\Delta }_2$-duality or the ${\Delta }_3$-duality of one-parameter family of frontals among three pseudo-spheres leads to the fact that the one-parameter family of frontals that are ${\Delta }_2$-duality or ${\Delta }_3$-duality each other share the same envelope. In addition, the hyperbolic and de Sitter parallels of the one-parameter family of spacelike frontals are also defined, and the existence conditions of the envelopes of such parallels are investigated correspondingly. Finally, two examples are provided to understand the theoretical results.