Study shape operator of relatively parallel hypersurfaces in n-dimensional geometry.
problem Characterize geometric properties of hypersurfaces in relative differential geometry.
method Analyze shape operator, principal curvatures, mean curvature, and affine normalization.
result Developed methods to study geometric properties of relatively parallel hypersurfaces.
The paper proves Bonnet-type theorems for hypersurfaces in 4D space.
problem Characterizing hypersurfaces in 4D space with constant mean curvatures.
method Using relative differential geometry and properties of relative mean curvatures.
result Proves Bonnet-type theorems for hypersurfaces with constant relative mean curvatures.
Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
The paper classifies hypersurfaces with parallel almost paracontact structures.
problem Classifying hypersurfaces with specific paracontact structures.
method Analyzing real affine hypersurfaces with a transversal vector field and studying the induced almost paracontact structures.
result Classification of hypersurfaces with parallel almost paracontact structures.
In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space Rn+1. Considering a relative normalization yˉ of an hypersurface Φ we decompose the corresponding Tchebychev vector Tˉ in two components, one parallel to the Tchebychev vector $\bar…
Classifies and describes hypersurfaces in Siklos spacetimes.
problem Characterizing hypersurfaces in Siklos spacetimes.
method Classification and description of totally geodesic and parallel hypersurfaces.
result A large class of minimal hypersurfaces is described.
Study classifies hypersurfaces with parallel structure Jacobi operator in complex quadric.
problem Characterizing real hypersurfaces with specific geometric properties.
method Defined and classified real hypersurfaces in complex quadric with parallel structure Jacobi operator.
result Complete classification of real hypersurfaces with parallel structure Jacobi operator.
Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.
problem Geometric and topological constraints on codimension-two spacelike submanifolds.
method Analysis of submanifolds with umbilical lightlike normal directions, using geometric and topological constraints.
result Any such submanifold is contained in a lightlike hypersurface, which is totally umbilical if the lightlike normal direction is umbilical.
Complete classification of centroaffine hypersurfaces with parallel cubic form.
problem Characterizing centroaffine hypersurfaces with specific geometric properties.
method Analyzing hypersurfaces with respect to the Levi-Civita connection of the centroaffine metric.
result A complete classification of locally strongly convex centroaffine hypersurfaces with parallel cubic form.
Classifies Calabi hypersurfaces with parallel Fubini-Pick form.
problem Classifying Calabi hypersurfaces with specific geometric properties.
method Classification based on parallel Fubini-Pick form and Levi-Civita connection.
result Classification of 2 and 3-dimensional Calabi hypersurfaces.
Classifies special hypersurfaces in Gödel spacetimes.
problem Characterizing hypersurfaces in Gödel spacetimes.
method Classification of parallel and totally geodesic hypersurfaces.
result Identified specific types of hypersurfaces in Gödel spacetimes.
Classifies special hypersurfaces in specific types of manifolds.
problem Characterizing geometric structures in non-reductive homogeneous spaces.
method Classification of parallel and totally geodesic hypersurfaces.
result Found a complete classification of these hypersurfaces.
Paper proves non-existence of certain hypersurfaces in complex quadric.
problem Non-existence of Hopf real hypersurfaces with parallel normal Jacobi operator.
method Introducing C-parallel and Reeb parallel normal Jacobi operators, proving non-existence theorems. result Non-existence of Hopf real hypersurfaces with C-parallel normal Jacobi operator. Study classifies real hypersurfaces with special Jacobi operator in complex hyperbolic quadric.
problem Characterizing real hypersurfaces with specific properties in complex hyperbolic quadric.
method Introduced Reeb parallel structure Jacobi operator and classified real hypersurfaces.
result Classification of real hypersurfaces with Reeb parallel structure Jacobi operator.
The study finds necessary and sufficient conditions for C1-hypersurfaces to have nowhere C1-regular parallel sets.
problem Conditions for C1-hypersurfaces to have nowhere C1-regular parallel sets. method Proves a necessary and sufficient condition for C1-hypersurfaces to have nowhere C1-regular parallel sets. result A necessary and sufficient condition for C1-hypersurfaces to have nowhere C1-regular parallel sets. This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose ∗-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel ∗-Ricci tensor in case…
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
problem Classifying hypersurfaces in the Sol4_0 geometry.
method Analyzing hypersurfaces with Codazzi tensors and parallel second fundamental forms.
result Full classification of hypersurfaces in Sol4_0, including parallel and totally umbilical types.
No real hypersurfaces found in complex Grassmannians with specific Jacobi operators.
problem Existence of real hypersurfaces in complex Grassmannians with semi-parallel structure Jacobi operator.
method Proved nonexistence through mathematical analysis.
result No real hypersurfaces exist in complex Grassmannians of rank two with semi-parallel structure Jacobi operator.
Mean curvature flow by parallel hypersurfaces is solved for isoparametric hypersurfaces.
problem Finding solutions to mean curvature flow by parallel hypersurfaces.
method Solving an ordinary differential equation for isoparametric hypersurfaces.
result Explicit solutions and exact collapsing time for isoparametric hypersurfaces of the sphere.
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
Paper classifies special Euclidean hypersurfaces with specific geometric properties.
problem Classifying Euclidean hypersurfaces with semi-parallel Moebius second fundamental form.
method Complete classification of hypersurfaces with three distinct principal curvatures.
result Classification of Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form.
The study characterizes canal hypersurfaces formed by non-null curves with parallel frame in Minkowski space-time.
problem Characterizing canal hypersurfaces in Minkowski space-time.
method Obtained canal hypersurfaces by pseudo hyperspheres or pseudo hyperbolic hyperspheres with parallel timelike normal vector field.
result Gaussian curvature, mean curvature, and principal curvatures of canal hypersurfaces were derived.
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians G2(Cm+2). Among them, Suh classified Hopf hypersurfaces M in G2(Cm+2) with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of gene…
This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.
We study three dimensional real hypersurfaces in CP^2 and CH^2 equipped with xi-parallel structure Jacobi operator. We prove that they are Hopf hypersurfaces and if additional α=0, we classify them.
We classify real hypersurfaces in CP^2and CH^2 equipped with pseudo-parallel structure Jacobi operator.
Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.
problem Existence of lightlike hypersurfaces in indefinite Sasakian manifolds.
method Proved non-existence through properties of second fundamental forms and induced structural tensors.
result No lightlike hypersurfaces can have parallel or recurrent second fundamental forms or induced structural tensors.
The paper classifies real hypersurfaces with a special structure in complex quadrics.
problem Characterizing real hypersurfaces with specific geometric properties in complex quadrics.
method Derived formulas for structure Jacobi operator and its derivative, classified hypersurfaces with Reeb parallel structure.
result Complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator.
The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassm…
Formula for spacelike submanifolds in warped products.
problem Finding formulas for spacelike submanifolds in semi-Riemannian warped products.
method Extending Simons' type formulas for spacelike submanifolds in semi-Riemannian warped products.
result Compact spacelike hypersurfaces with parallel mean curvature and non-negative sectional curvature are isoparametric hypersurfaces.
We prove the non-existence of real hypersurfaces in CP^2 and CH^2 whose structure Jacobi operator is Lie D-parallel.
In 1970, Samuel I. Goldberg and Kentaro Yano defined the notion of noninvariant hypersurface of a Sasakian manifold [1]. In this paper we have studied the properties of parallel vector fields with respect to induced connection on the noninvariant hypersurface M of a Sasakian manifold M~ with $(φ, g, u, v, λ)-…
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
The paper characterizes hypersurfaces in curved spaces using their geometry.
problem Geometric characterization of hypersurfaces in curved spaces.
method Extrinsic geometry analysis of conformally and radially flat hypersurfaces.
result Classification of hypersurfaces in terms of rotation and semi-parallel hypersurfaces.
The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.
problem Stability of CMC hypersurfaces with free boundaries.
method Analysis and numerical computations.
result Equilibrium hypersurfaces are stable without self-intersection in all dimensions.
In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians G2(Cm+2). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in G2(Cm+2).
Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.
problem Characterizing totally umbilical hypersurfaces in Spin^c manifolds with specific spinor fields.
method Proving constant mean curvature for hypersurfaces carrying parallel, real or imaginary Killing spinors.
result Results extend to Spin^c case, generalizing O. Kowalski's theorem.
In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S1m+1 with parallel para-Blaschke tensors.
The non-existence of three dimensional real hypersurfaces in non-flat complex space forms with parallel *-Ricci tensor is proved.At the end of the papaer ideas for further research on *-Ricci tensor are provided.
The paper studies inverse mean curvature flow on hypersurfaces in space forms.
problem Analyzing the inverse mean curvature flow on hypersurfaces in space forms.
method Investigates the existence and properties of the flow for isoparametric hypersurfaces.
result Characterizes the flow and solutions explicitly under certain conditions.
We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …
Two Spin^c structures characterize hypersurfaces in product spaces.
problem Characterizing hypersurfaces in product spaces via Spin^c structures.
method Restricting parallel spinor fields to hypersurfaces.
result Totally umbilical hypersurfaces have constant mean curvature.
The paper studies f-stability of hypersurfaces in gradient Ricci solitons.
problem Estimating the f-stability index of constant weighted mean curvature hypersurfaces. method Analyzes hypersurfaces in shrinking gradient Ricci solitons with parallel fields.
result Provides an estimate for the f-stability index and necessary conditions for equality. Study of superintegrable systems linked to affine hypersurfaces.
problem Understanding superintegrable systems through geometric structures.
method Established a correspondence between superintegrable systems and affine hypersurfaces, defining conformal equivalence.
result Identified conformal classes of abundant manifolds with abundant hypersurface immersions.
The paper defines and studies equiaffine isoparametric hypersurfaces and functions.
problem Understanding and characterizing equiaffine isoparametric hypersurfaces and functions.
method Introduced equiaffine parallel hypersurfaces, defined equiaffine isoparametric hypersurfaces and functions, and proved their relationship.
result Equiaffine isoparametric hypersurfaces are regular level sets of equiaffine isoparametric functions.
In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space S1m+1, to cover the conformal space Q1m+1, so that the conformal geometry of regular space-like hypersurfaces in Q1m+1 is treated as that of hypersurfaces in ${\mathbb S}…
The study classifies special Lorentz surfaces in a 4-space with neutral metric.
problem Classifying meridian surfaces in a pseudo-Euclidean 4-space.
method Constructing and classifying meridian surfaces with specific properties.
result There exist meridian surfaces with parallel normalized mean curvature vector field but not parallel mean curvature vector.