This paper investigates the concept of cracking and overturning to analyze the impact of local density perturbations on the stability of self-gravitating compact objects in the framework of $$f(R,\phi , X)$$f ( R , ϕ , X ) theory of gravity, where R, $$\phi $$ϕ , and X denote the Ricci scalar, scalar potential, and kinetic term, respectively. In this context, we developed the hydrostatic equilibrium equation for spherically symmetric spacetime with anisotropic matter configuration and subsequently employed the Krori Barua technique. We then perturb the hydrostatic equilibrium state of the configuration by employing the local density perturbation technique, while taking into account the barotropic equation of state. To validate this technique, we employed it on different compact stars namely, Her X-1, SAX J1808.4-3658, 4U 1820-30, PSR J1614-2230, Vela X-1, and Cen X-3, and found that all stars exhibit cracking or overturning for a specific range of model parameters. Conclusively, this study emphasizes that the proposed cracking technique provides significant insights into the stability analysis of self-gravitating compact objects.

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form ${\Omega }^n=N_T^l{\times }_f{N}_{\varphi }^k$ in a complex projective space $\mathbb{C}{P}^{2m}\left(4\right)$. Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function $\mathbb{E}\left(f\right)$, and first non-zero eigenvalue ${\lambda }_1$ to prove that stable l-currents do not exist and also that the homology groups have vanished in ${\Omega }^n$. As an application of the non-existence of the stable currents in ${\Omega }^n$, we show that the fundamental group ${\pi }_1\left({\Omega }^n\right)$ is trivial and ${\Omega }^n$ is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

Recently, B.-Y. Chen and O.J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. In this paper, first we study pointwise slant and pointwise pseudo-slant submanifolds of almost contact metric manifolds and then using this notion, we show that there exist a non-trivial class of warped product pointwise pseudo-slant submanifolds of Sasakian manifolds by giving some useful results, including a characterization.

The aim of this work is to study the general formalism and applications of polytropic stars within the framework of a well-famed modified theory of gravity namely $$\mathbb {F}(R)$$ F ( R ) theory. To accomplish this task, we assume the static spherically symmetric and Schwarzschild spacetimes, respectively, for defining the star’s interior and exterior geometries. By applying two polytropic equations of state, we derive physically well-behaved solutions to $$\mathbb {F}(R)$$ F ( R ) field equations. Firstly, we develop the hydrostatic equilibrium and Lane–Emden equations for the isotropic fluid case within this framework of gravity. Next, we analyze the impact of isotropic pressure on stellar structures and graphically assess the physical behavior of isotropic polytropes using energy conditions and stability criteria. Additionally, the physical acceptability conditions are examined for the proposed solutions. The results indicate that this theory produces more viable and stable polytropes as compared to general relativity.