We prove that if an η -Einstein para-Kenmotsu manifold admits a conformal η -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η -Ricci soliton is Einstein if its potential vector field V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η -Ricci soliton and satisfy our results. We also have studied conformal η -Ricci soliton in three-dimensional para-cosymplectic manifolds.

In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.

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.