arXiv research

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

Classifies surfaces with special curvature properties.

The study examines principal directions and curvatures of Lagrangian submanifolds.

The paper constructs all cmc hypersurfaces with two principal curvatures.

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…

The study of submanifolds with constant principal curvatures in symmetric spaces.

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

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

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

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

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 …

The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.

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…

Study on hypersurfaces with specific curvature conditions.

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distin…

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

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…

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

Study on $G_{2}$ actions on symmetric spaces, focusing on orbit properties.

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that …

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

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

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

The paper classifies biconservative hypersurfaces in 4D space forms.

The paper studies mean curvature flows on specific orbits of Hermann actions.

Study bifurcations of curves on surfaces in Minkowski 3-space.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

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 …

The paper characterizes surfaces using invariants.

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…

Classifies hypersurfaces with specific curvature properties in 4D space.

Study curvature of determinant bundle over Teichmüller space.

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersur…

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…

The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.

The paper classifies and explores isoparametric hypersurfaces in pseudo-Riemannian space forms.

New examples of hypersurfaces found in quaternionic hyperbolic spaces.

Classifies surfaces in specific 3-manifolds with isometry group.

Classifies surfaces in Euclidean space with specific curvature relations.

The paper classifies hypersurfaces in product spaces with specific curvature properties and finds that only rotational ones admit almost Ricci solitons.

Author reduces the Minkowski problem to the problem of construction the G-deformations preserving the product of principal curvatures for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The contin…

In this paper, we study biconservative hypersurfaces of index 2 in $\mathbb E^{5}_{2}$. We give the complete classification of biconservative hypersurfaces with diagonalizable shape operator at exactly three distinct principal curvatures. We also give an explicit example of biconservative hypersurfaces with four distin…

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathb…

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

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…

New pinching estimates control curvature ratios in inverse curvature flows.

The study classifies biharmonic real hypersurfaces in complex projective spaces.

The study proves curvature bounds for quotient spaces of isometric actions.