arXiv research

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.

Study on discrete surfaces with constant principal curvature for nanocarbon applications.

The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

The study examines spacelike hypersurfaces in Minkowski space with constant $σ_{n-1}$ curvature.

The paper classifies various types of hypersurfaces in a product space.

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…

We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.

The study classifies hypersurfaces with constant principal curvatures in $\mathbb{S}^3 imes \mathbb{R}$ and $\mathbb{H}^3 imes \mathbb{R}$.

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show …

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given i…

We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. …

The paper classifies surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.

In this paper, we have studied biharmonic hypersurfaces in space form $\bar{M}^{n+1}(c)$ with constant sectional curvature $c$. We have obtained that biharmonic hypersurfaces $M^{n}$ with at most three distinct principal curvatures in $\bar{M}^{n+1}(c)$ has constant mean curvature. We also obtain the full classificatio…

Compact Dupin hypersurfaces without constant Lie curvatures found.

The paper studies Einstein hypersurfaces in a specific warped product space.

Study proves surfaces with constant curvature are simple shapes.

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatu…

In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: is every hypersurface with constant principal curvatures isoparametric?

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, w…

Classifies isoparametric hypersurfaces in 3D manifolds.

Paper corrects a proof about biharmonic hypersurfaces with three distinct curvatures.

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to …

Paper finds conditions for minimal hypersurfaces in S^6 with constant scalar curvature.

Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared…

Novel symmetry found in nanocarbons' discrete principal curvature structure.

Study finds all helical surfaces with a constant ratio of principal curvatures.

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.

In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove…

The paper examines biconservative hypersurfaces with constant curvature in space forms.

We study surfaces in $\R^4$ whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We …

In this paper, we obtain some properties of biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with at most five distinct prin…

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

We study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $aκ_1+bκ_2=c$, where $κ_i$ are the principal curvatures, and $a,b$ and $c$ are constant.

Paper proves existence of weighted constant scalar curvature metrics.

Planes and spheres are the only stationary surfaces with constant Gauss curvature.

Lie minimal surfaces are characterized by differential equations of principal curvatures.

New examples of hypersurfaces found in quaternionic hyperbolic spaces.

The paper classifies isoparametric hypersurfaces in product spaces with different curvatures.

The paper constructs all cmc hypersurfaces with two principal curvatures.

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subje…

Complete classification of isoparametric hypersurfaces in product spaces of space forms.

Lie sphere geometry helps classify Dupin hypersurfaces.

Study on null hypersurfaces with constant angle in Lorentzian manifolds.

The study characterizes Clifford hypersurfaces in terms of curvature constants.