In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the $\eta$-Ricci–Yamabe soliton ($\eta$-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an $\eta$-RY soliton, an $\eta$-Ricci soliton, and an $\eta$-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an $\eta$-RY soliton, an $\eta$-Ricci soliton, an $\eta$-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field $\omega$ of the soliton is of gradient type =:grad$(\gamma )$ and provide some examples of an $\eta$-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.

In recent years, the modeling of compact astrophysical objects (COs) has garnered significant attention from various research groups, particularly in efforts to determine their stable structures. This interest has been further amplified by the incorporation of dark energy as an additional source within the relativistic interior geometries of these stellar objects. In this paper, we present new structural properties of anisotropic, static, and spherically symmetric compact stars, characterized by a two-fluid distribution comprising ordinary baryonic matter and dark energy, within the framework of the gravity theory f(R). We derive a novel class of exact analytical solutions to the modified field equations by employing the well-established Tolman–Buchdahl solutions as seed ansatz for the $$g_{tt}$$ g tt and $$g_{rr}$$ g rr metric potentials, in conjunction with a linear dark energy equation of state. The unknown parameters involved in these seed solutions, along with the dark energy coupling factor $$\omega $$ ω , are determined by a smooth matching of the interior and exterior regions in the hypersurface of the boundary. To analyze the physical viability of our model, we apply it to the compact star PSR $$J1614-2230$$ J 1614 - 2230 , using the widely studied and cosmologically consistent $$f(R)=R+\chi \Big (R^{2}+\eta R^{3}\Big )$$ f ( R ) = R + χ ( R 2 + η R 3 ) gravity model. The obtained space-time geometry is assessed on the basis of several physical constraints, including the regularity of metric components, the viability of matter variables, the validity of state parameters and energy conditions, and various stability factors. Additionally, we examine the mass-radius profile, compactness, and surface redshift to ensure the model’s physical acceptability. Notably, our analysis reveals that the maximum allowable mass and compactness of the proposed dark energy star model exceed observational data, providing strong implications within this modified gravity framework. This suggests that such a model has the potential to surpass conventional observational predictions. In conclusion, our results confirm that the proposed solutions are physically viable and realistic, effectively mimicking a stable ultra-compact dark-energy star. These findings offer new insights into the relativistic stellar framework, highlighting the intricate interplay between extended gravity theories and a two-fluid distribution.

In this study, the authors focus on quasi-hemi-slant submanifolds ($qhs$-submanifolds) of $(\alpha ,\beta )$-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory.

A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of $\eta$-Ricci solitons ($\eta$-RS) for an interesting manifold called the $(\epsilon )$-Kenmotsu manifold ($(\epsilon )$-$KM$), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and $\eta$-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient $\eta$-Ricci solitons (a special case of $\eta$-Ricci soliton), with a Poisson equation on the same ambient manifold for a SSM-connection. In addition, we derive an inequality for the lower bound of gradient $\eta$-Ricci solitons for $(\epsilon )$-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the $(\epsilon )$-Kenmotsu manifold equipped with a semi-symmetric metric connection.

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions.