Let $(M,g)$ be an $n$-dimensional (pseudo-)Riemannian manifold and $TM$ be its tangent bundle $TM$ equipped with the complete lift metric ${}^{C}g$. First, we define a Ricci quarter-symmetric metric connection $\overline{\mathrm{\nabla }}$ on the tangent bundle $TM$ equipped with the complete lift metric ${}^{C}g$. Second, we compute all forms of the curvature tensors of $\overline{\mathrm{\nabla }}$ and study their properties. We also define the mean connection of $\overline{\mathrm{\nabla }}$. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $TM$ to become a Ricci soliton and a gradient Ricci soliton concerning $\overline{\mathrm{\nabla }}$ are presented. Finally, we search conditions for the tangent bundle $TM$ to be locally conformally flat with respect to $\overline{\mathrm{\nabla }}$.

In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space ${\mathbb{E}}_3^7$. These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures effectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and offer a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a differential operator that characterizes the geometry of the hypersurfaces, and a specific $7\times 7$ matrix denoted as $\mathcal{A}$. By studying this relation, we gain further insights into the geometric structure and differential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in ${\mathbb{E}}_3^7$, offering mathematical insights and establishing connections between various geometric quantities and operators associated with this family.

In this study, we introduce partner ruled surfaces according to the Flc frame that is defined on a polynomial curve. First, the conditions of each couple of two partner ruled surfaces to be simultaneously developable and minimal are investigated. Then, the asymptotic, geodesic and curvature lines of the parameter curves of the partner ruled surfaces are simultaneously characterized. Finally, the examples of the partner ruled surfaces are given, and their graphs are drawn.

We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair $(G,B)$, which consists of a cuspidal edge $G$ in ${\mathbb{R}}^3$ that contains a distinguishing regular curve $B$, are classified. This classification was used as a means to investigate the contact that a general cuspidal edge $G$ equipped with a regular curve $B\subset G$ has with planes. The singularities of the height functions on $(G,B)$ are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge.

Given an immersed hypersurface $M^n$ in the Euclidean space $E^{n+1}$, the tangential component $\mathit{\omega }$ of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function $\sigma$ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface $M^n$ in $E^{n+1}$ of positive Ricci curvature with shape operator $T$ invariant under $\mathit{\omega }$ and the support function $\sigma$ satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface $M^n$ in $E^{n+1}$ with the gradient of support function $\sigma$, an eigenvector of the shape operator $T$ with eigenvalue function the mean curvature $H$, and the integral of the squared length of the gradient $\mathrm{\nabla }\sigma$ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $M^n$ of positive Ricci curvature in $E^{n+1}$ has an incompressible basic vector field $\mathit{\omega }$, if and only if $M^n$ is isometric to a sphere.