Search Results (17)

In this paper, we deal with the canal hypersurfaces that are formed as the envelope of a family of pseudo-hyperspheres or pseudo-hyperbolic hyperspheres with centers lying on a pseudo-null curve with Bishop vector fields in four-dimensional Lorentz–Minkowski space. We give main theorems which contain the parametric expressions of these canal hypersurfaces along with their Gaussian, mean, and principal curvatures and important geometric characterizations. We also provide these characterizations for tubular hypersurfaces. Finally, we construct an example to allow for better understanding and comprehension of the results. Full article

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space ${\mathbb{E}}^n$, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When $(n-1)$-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space ${\mathbb{E}}_1^4$ and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in ${\mathbb{E}}_1^4$, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries. Full article

This study explores parallel hypersurfaces in four-dimensional Euclidean space ${\mathbb{E}}^4$, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry. Full article

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the gradient r-ANRY soliton. We also exhibit a Schur-type inequality and discuss the triviality of the gradient r-ANRY soliton in the case of a compact manifold. Finally, we demonstrate the completeness and noncompactness of the r-Newton-Ricci-Yamabe soliton on the hypersurface of the Riemannian manifold. Full article

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form $(\overline{M},\overline{g})$, where $\overline{M}=\mathbb{R}{\times }_{f}M$ and $\overline{g}=ϵd{t}^2+f^2(t)g_M$. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where $(\overline{M},\overline{g})$ is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ${\partial }_t$ and the so-called support function $\theta$ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below. Full article

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored. Full article

The twisted hypersurfaces $\mathfrak{x}$ with the $(0,0,0,0,1)$ rotating axis in five-dimensional Euclidean space ${\mathbb{E}}^5$ is considered. The fundamental forms, the Gauss map, and the shape operator of $\mathfrak{x}$ are calculated. In ${\mathbb{E}}^5$, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces $\mathfrak{x}$ are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of $\mathfrak{x}$ are determined. Additionally, the Laplace–Beltrami operator relation of $\mathfrak{x}$ is given. Full article

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature. Full article

We obtain some properties of a hyperbolic Ricci soliton with certain types of potential vector fields, and we point out some conditions when it reduces to a trivial Ricci soliton. We also study those soliton submanifolds whose vector fields are the tangential components of a concurrent vector field on the ambient manifold, and in particular, we show that a totally umbilical hyperbolic Ricci soliton is an Einstein manifold. We prove that if the hyperbolic Ricci soliton hypersurface of a Riemannian manifold of constant curvature and endowed with a concurrent vector field has a parallel shape operator, then it is a metallic-shaped hypersurface, and we determine some conditions for it to be minimal. Moreover, we show that it is also a pseudosymmetric hypersurface. Full article

We develop an action principle to construct the field equations for dissipative/resistive general relativistic two-temperature plasmas, including a neutrally charged component. The total action is a combination of four pieces: an action for a multifluid/plasma system with dissipation/resistivity and entrainment; the Maxwell action for the electromagnetic field; the Coulomb action with a minimal coupling of the four-potential to the charged fluxes; and the Einstein–Hilbert action for gravity (with the metric being minimally coupled to the other action pieces). We use a pull-back formalism from spacetime to abstract matter spaces to build unconstrained variations for the neutral, positively, and negatively charged fluid species and for three associated entropy flows. The full suite of field equations is recast in the so-called “$3+1$” form (suitable for numerical simulations), where spacetime is broken up into a foliation of spacelike hypersurfaces and a prescribed “flow-of-time”. A previously constructed phenomenological model for the resistivity is updated to include the modified heat flow and the presence of a neutrally charged species. We impose baryon number and charge conservation as well as the Second Law of Thermodynamics in order to constrain the number of free parameters in the resistivity. Finally, we take the Newtonian limit of the “$3+1$” form of the field equations, which can be compared to existing non-relativistic formulations. Applications include main sequence stars, neutron star interiors, accretion disks, and the early universe. Full article

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with $k=2,3$ distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. Full article

Minimal compact hypersurface in the unit sphere ${\mathbb{S}}^{n+1}$ having squared length of shape operator ${∥A∥}^2<n$ are totally geodesic and with ${∥A∥}^2=n$ are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in ${\mathbb{S}}^{n+1}$. One finds a naturally induced vector field w called the associated vector field and a smooth function $\rho$ called support function on the hypersurface M of ${\mathbb{S}}^{n+1}$. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in ${\mathbb{S}}^5$ to be totally geodesic is that the support function $\rho$ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in ${\mathbb{S}}^{n+1}$, ($n>2$), provided the scalar curvature $\tau$ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in ${\mathbb{S}}^{n+1}$ is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field $Aw$. Full article

Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the directions of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note, in this paper, that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry. In particular, we show that a geometric origin for dark energy can be proposed within this framework. Full article

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere $S^{2n+1}$ with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere $S^{2n+1}$ . Full article

In this note, we give a characterization of a class of minimal translation graphs generated by planar curves. Precisely, we prove that a hypersurface that can be written as the sum of n planar curves is either a hyperplane or a cylinder on the generalized Scherk surface. This result can be considered as a generalization of the results on minimal translation hypersurfaces due to Dillen–Verstraelen–Zafindratafa in 1991 and minimal translation surfaces due to Liu–Yu in 2013. Full article