The present paper is to deliberate the class of $ \epsilon $-Kenmotsu manifolds which admits conformal $ \eta $-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal $ \eta $-Ricci soliton of $ \epsilon $-Kenmotsu manifolds. Moving further, we investigate some curvature conditions admitting conformal $ \eta $-Ricci solitons on $ \epsilon $-Kenmotsu manifolds. Next, we consider gradient conformal $ \eta $-Ricci solitons and we present a characterization of the potential function. Finally, we develop an illustrative example for the existence of conformal $ \eta $-Ricci soliton on $ \epsilon $-Kenmotsu manifold.

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.

In this study, the ruled developable surfaces with pointwise 1-type Gauss map of Frenet-type framed base (Ftfb) curve are introduced in Euclidean 3-space. The tangent developable surfaces, focal developable surfaces, and rectifying developable surfaces with singular points are considered. Then the conditions for the Gauss map of these surfaces to be pointwise 1-type are obtained separately. In order to form a basis for the study, first, the basic concepts related to the Ftfb curve and the Gauss map of a surface are recalled. Later, the necessary and sufficient conditions are found for these surfaces to be of the pointwise 1-type of the Gauss map. Finally, examples for each type of these surfaces are given, and their graphics are illustrated.

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.