Classification Of Hypersurfaces Research Articles

Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\). Next, we give a complete proof of non-existence of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with shape operator of Codazzi type. Motivated by this result, we give a complete classification of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with Reeb parallel shape operator.

A classification of biconservative hypersurfaces in a pseudo-Euclidean space

In this paper, we study biconservative hypersurfaces of index 2 in E25. We give the complete classification of biconservative hypersurfaces with diagonalizable shape operator with exactly three distinct principal curvatures. We also give an explicit example of biconservative hypersurfaces with four distinct principal curvatures.

Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection

The cubic form of a hypersurface in the affine space is an important invariant in affine differential geometry. It is known that many interesting classes of hypersurfaces in the affine space are characterized by the cubic form. For example, in equiaffine differential geometry, quadrics are characterized as nondegenerate hypersurfaces with vanishing cubic form, which is known as theorem of Pick–Berwald. In this paper, we shall give some classification results for nondegenerate centroaffine surfaces in terms of the cubic form and the induced connection. First, we prove that a nondegenerate centroaffine surface whose cubic form is parallel relative to the induced connection is a piece of an ellipsoid or a hyperboloid centered at the origin. Next, we prove that a nondegenerate centroaffine surface whose traceless part of the cubic form is recurrent relative to the induced connection is a piece of a certain centroaffine minimal ruled surface such that the Pick function vanishes and the curvature of the centroaffine metric equals to 1.

Complete classification of biconservative hypersurfaces with diagonalizable shape operator in the Minkowski 4-space

In this paper, we study biconservative hypersurfaces in the four-dimensional Minkowski space [Formula: see text]. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in [Formula: see text].

Real hypersurfaces in the complex quadric with parallel normal Jacobi operator

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.

Isoparametric hypersurfaces in Minkowski spaces

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes, Minkowski hyperspheres and F⁎-Minkowski cylinders in a Minkowski space with BH-volume (resp. HT-volume) form are all isoparametric hypersurfaces with one and two distinct constant principal curvatures respectively. Moreover, we give a complete classification of isoparametric hypersurfaces in Randers–Minkowski spaces.

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb invariant Ricci tensor

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m≥2 from the equation of Gauss. Next we derive a new formula for the Ricci tensor S of M in SU2,m/S(U2⋅Um). Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um) with Reeb invariant Ricci tensor, that is, LξS=0. Each can be described as a tube over a totally geodesic SU2,m−1/S(U2⋅Um−1) in SU2,m/S(U2⋅Um) or a horosphere whose center at infinity is singular.

Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal or $\frak A$-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with commuting Ricci tensor.

On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1

In this paper, we study isometrically immersed hypersurfaces of the Euclidean space En+1 satisfying the condition LrH r+i = λHr+1for an integer r ( 0 ≤ r ≤ n — 1), where Hr+iis the (r + 1)th mean curvature vector field on the hypersurface, Lr is the linearized operator of the first variation of the (r + 1) th mean curvature of hypersurface arising from its normal variations. Having assumed that on a hypersurface x : Mn → En+1, the vector field Hr+i be an eigenvector of the operator Lr with a constant real eigenvalue λ, we show that, Mn has to be an Lr-biharmonic, Lr-1-type, or Lr-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of hypersurfaces, named the weakly convex hypersurfaces (i.e. on which principal curvatures are nonnegative). We prove that, any weakly convex Euclidean hypersurface satisfying the condition Lr Hr+i = λ Hr+i for an integer r ( 0 ≤ r ≤ n — 1), has constant mean curvature of order (r + 1). As an interesting result, we have that, the Lr-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality.

Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians with Commuting Structure Jacobi Operators

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, \({R_{\xi} \phi T = TR_{\xi} \phi}\), where T = A or T = S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. Using simultaneous diagonalization for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition, respectively.

Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

A classification of biharmonic hypersurfaces in the Minkowski spaces of arbitrary dimension

A classification of biharmonic hypersurfaces in the Minkowski spaces of arbitrary dimension

A Class of Special Hypersurfaces in Real Space Forms

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

Lorentzian Submanifolds in Semi-Euclidean Spaces with Pointwise 1-Type Gauss Map

In this work first, we survey the most recent classification results for submanifolds with pointwise 1-type Gauss map. Then, we study a class of hypersurfaces with vanishing Gauss-Kronecker curvature in terms of type of their Gauss map.

Four-Dimensional Projective Orbifold Hypersurfaces

ABSTRACTWe classify four-dimensional quasismooth weighted hypersurfaces with small canonical class and verify a conjecture of Johnson and Kollár on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi–Yau and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.

Classification of hypersurfaces with constant Möbius Ricci curvature in $\mathbb{R}^{n+1}$

Let $f: M^n\rightarrow \mathbb{R}^{n+1}$ be an immersed umbilic-free hypersurface in an $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ with standard metric $I=df\cdot df$. Let $II$ be the second fundamental form of the hypersurface $f$. One can define the Mobius metric $g=\frac{n}{n-1}(\|II\|^2-n|{\rm tr}II|^2)I$ on $f$ which is invariant under the conformal transformations (or the Mobius transformations) of $\mathbb{R}^{n+1}$. The sectional curvature, Ricci curvature with respect to the Mobius metric $g$ is called Mobius sectional curvature, Mobius Ricci curvature, respectively. The purpose of this paper is to classify hypersurfaces with constant Mobius Ricci curvature.

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.

REAL HYPERSURFACES OF NON-FLAT COMPLEX SPACE FORMS WITH GENERALIZED ξ-PARALLEL JACOBI STRUCTURE OPERATOR

AbstractThe aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space TpM of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξl)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξl = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.

A filtration for isoparametric hypersurfaces in Riemannian manifolds

Motivated by the well known Cartan-Munzner equations and the Chern conjecture, we introduce the notion of k-isoparametric hypersurface in an (n + 1)- dimensional Riemannian manifold for k = 0,1,...,n. Many fundamental and inter- esting results ( towards the classification of homogeneous hypersurfaces among other things ) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces. Further investigations in this direction are expected.

Semi-parallel symmetric operators for Hopf hypersurfaces in complex two-plane Grassmannians

In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in \(G_2({\mathbb C}^{m+2})\).