Classification Of Hypersurfaces Research Articles

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

AbstractWe study hypersurfaces of the four‐dimensional Thurston geometry , which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.

Screen Semi-Invariant Lightlike Hypersurfaces on Hermite-Like Manifolds

Hermite-like manifolds, which admit two different, almost complex structures, can be considered a general concept of Hermitian manifolds. Factoring in the effects of these two complex structures on the radical, screen, and transversal spaces, a new classification of lightlike hypersurfaces of Hermite-like manifolds is proposed in the present paper. Moreover, an example of screen semi-invariant lightlike hypersurfaces of Hermite-like manifolds is provided. Besides, some results on these hypersurfaces admitting a statistical structure are obtained. Further, screen semi-invariant lightlike hypersurfaces are investigated on Kaehler-like statistical manifolds. In addition, several characteristics of totally geodesic, mixed geodesic, and totally umbilical screen lightlike hypersurfaces are obtained. Finally, the need for further research is discussed.

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

In the unit Euclidean sphere Sn+1, we deal with a class of hypersurfaces that were characterized in [23] as the critical points of a variational problem, the so-called (r, s)-linear Weingarten hypersurfaces (0 ≤ r ≤s ≤ n−1); namely, the hypersurfaces of Sn+1 that has a linear combination arHr+1+...+asHs+1 of their higher order mean curvatures Hr+1 and Hs+1 being a real constant, where ar,..., ar are nonnegative real numbers (with at least one non zero). By assuming a geometric constraint involving the higher order mean curvatures of these hypersurfaces, we prove a uniqueness result for strongly stable (r, s)-linear Weingarten hypersurfaces immersed in a certain region determined by a geodesic sphere of Sn+1. We also establish a nonexistence result in another region of Sn+1 for strongly stable Weingarten (r, s)-linear hypersurfaces.

Classification of semi-parallel hypersurfaces of the product of two spheres

It is known that Mendonça and Tojeiro (2013) [19] have established a complete classification of parallel submanifolds in the product manifold Qk1n1×Qk2n2, where Qk1n1 (resp. Qk2n2) is an n1-dimensional (resp. n2-dimensional) real space form with constant curvature k1 (resp. k2). In this paper, motivated by this result with considering further generalization, we study those semi-parallel hypersurfaces in case Qk1n1=Sk1n1 and Qk2n2=Sk2n2 with k1,k2>0. As the main result, we classify semi-parallel hypersurfaces of Sk1n1×Sk2n2 for n1,n2≥2.

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5), and textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1.

CYCLIC RICCI SEMISYMMETRIC REAL HYPERSURFACES IN THE COMPLEX QUADRIC

We introduce the notion of cyclic Ricci semisymmetric real hypersurfaces in the complex quadric Qm=SOm+2∕SOmSO2. We also give a classification of real hypersurfaces in the complex quadric Qm=SOm+2∕SOmSO2 with cyclic semisymmetric Ricci tensor.

The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields

&lt;abstract&gt;&lt;p&gt;Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n &amp;gt; 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.&lt;/p&gt;&lt;/abstract&gt;

The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor.

Einstein hypersurfaces in irreducible symmetric spaces

We give a full classification of Einstein hypersurfaces in irreducible Riemannian symmetric spaces of rank greater than 1 (the classification in the rank-one case was previously known). There are two classes of such hypersurfaces. The first class consists of codimension one Einstein solvmanifolds in irreducible symmetric spaces of noncompact type constructed via the Iwasawa decomposition. The second class consists of two exceptional families in low-dimensional symmetric spaces \({\overline{M}}=\text {SU}(3)/\text {SO}(3)\) and \({\overline{M}}=\text {SL}(3)/\text {SO}(3)\). Any Einstein hypersurface M in such space \({\overline{M}}\) is developable: it is foliated by totally geodesic spheres (respectively, by totally geodesic hyperbolic planes) of \({\overline{M}}\), with the space of leaves being a special Legendrian surface in \(S^5\) (respectively, a proper affine sphere in a \({\mathbb {R}}^3\)).

Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

AbstractFirst we introduce a new notion of generalized Killing Ricci tensor which is equivalent to the notion of cyclic parallel Ricci tensor for real hypersurfaces in the complex quadric . Next, in terms of ‐principal or ‐isotropic unit normal vector fields we give a complete classification of real hypersurfaces in with cyclic parallel Ricci tensor.

Isoparametric hypersurfaces in conic Finsler manifolds

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that there are probably other isoparametric hypersurfaces in conic Minkowski spaces besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in Kropina spaces with constant flag curvature.

Certain class of hypersurfaces in a Kenmotsu space forms

Certain class of hypersurfaces in a Kenmotsu space forms

The equivalence theory for infinite type hypersurfaces in ℂ²

We develop a classification theory for real-analytic hypersurfaces in C 2 \mathbb {C}^{2} in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in C 2 \mathbb {C}^{2} in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR-DS technique in CR-geometry.

Hypersurfaces with radial mean curvature in space forms

In this paper, we study two classes of hypersurfaces, namely, the DRMC-hypersurfaces and the HDRMC-hypersurfaces in space forms Mn+1(c), c=−1,0,1, these classes include the Weingarten hypersurfaces of the spherical type obtained in |10|. For n= 2, we present a way to obtain DRMC-surfaces and HDRMC-surfaces in M3(c) using two holomorphic functions. Also, we classify the DRMC-hypersurfaces of rotation in Mn+1(c) and the HDRMC-hypersurfaces of rotation in Rn+1.

A certain η-parallelism on real hypersurfaces in a nonflat complex space form

Abstract In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the η-parallelism of the tensor field h(= (1/2)𝓛 ξ ϕ). In addition we investigate real hypersurfaces whose tensor h is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor h is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting η-parallelism with respect to the tensor h.

Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in ℂ3

Abstract Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.

Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group S_d. We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

Real hypersurfaces in a nonflat complex space form whose ⁎-Ricci tensor is [formula omitted]-recurrent

In this paper, we give a classification of real hypersurfaces M2n−1 in a nonflat complex space form M˜n(c)(n≧2) whose ⁎-Ricci tensor is D-recurrent. By virtue of this result, we also know a classification of real hypersurfaces in a nonflat complex space form whose ⁎-Ricci tensor is D-parallel.