Hypersurfaces with isotropic para-Blaschke tensor

Let Mnbe an n-dimensional submanifold without umbilical points in the(n + 1)-dimensional unit sphere Sn+1.Four basic invariants of Mnunder the Moebius transformation group of Sn+1are a 1-form Φ called moebius form,a symmetric(0,2) tensor A called Blaschke tensor,a symmetric(0,2) tensor B called Moebius second fundamental form and a positive definite(0,2) tensor g called Moebius metric.A symmetric(0,2) tensor D = A + μB called para-Blaschke tensor,where μ is constant,is also an Moebius invariant.We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg.One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor.When λ is not constant,all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.     

Surfaces with isotropic Blaschke tensor in S 3

Let M2 be an umbilic-free surface in the unit sphere S3. Four basic invariants of M2 under the Moebius transformation group of S3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B and Moebius form Φ. We call the Blaschke tensor is isotropic if there exists a smooth function λ such that A = λg. In this paper, We classify all surfaces with isotropic Blaschke tensor in S3.     

Hypersurfaces with parallel para‐Laguerre tensor in

Let be an ‐dimensional hypersurface with vanishing Laguerre form in , be the Laguerre second fundamental form and be the Laguerre tensor of the immersion x. We define a symmetric (0, 2) tensor which is so‐called the para‐Laguerre tensor of x, where λ is a constant. If , we say that x is of parallel para‐Laguerre tensor, where ? is the Levi‐Civita connection of the Laguerre metric g. An eigenvalue of the para‐Laguerre tensor is called a para‐Laguerre eigenvalue of x. The aim of this paper is to classify all oriented hypersurfaces in with parallel para‐Laguerre tensor or with three distinct constant para‐Laguerre eigenvalues one of which is simple.     

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Möbius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {E_i}. Then we give a complete classification for spacelike Möbius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Möbius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Möbius form.     

The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

We study Lorentzian affine hypersurfaces in Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R⁴ is given.     

SUBMANIFOLDS IN S^n＋p WITH PARALLEL MOEBIUS FORM

In this paper,the rigidity theorems of the submanifolds in S^n p with parallel Moebius form and constant MObius scalar curvature are given.     

Blaschke’s problem for hypersurfaces

We solve Blaschke’s problem for hypersurfaces of dimension . Namely, we determine all pairs of Euclidean hypersurfaces that induce conformal metrics on M ⁿ and envelop a common sphere congruence in .     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

Ruled surfaces of Weingarten type in Minkowski 3-space, II

In our previous paper of the same title, we did not study the ruled surfaces of Weingarten type M : x(s, t)=α(s)+t β (s) in Minkowski 3-space with vector fields β and β′ along the base curve β such that β is nowhere null but β′ is null everywhere. We here fulfill our project by investigating this remaining case.     

A characterization of constant isotropic immersions by circles

Motivated by the well-known result of Nomizu and Yano [4], we provide a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. As an immediate consequence of this result, we characterize Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Received: 11 January 2002     

The classification of 4-dimensional homogeneous D'Atri spaces revisited

In this short note we correct the (incomplete) classification theorem from [F. Podestà, A. Spiro, Four-dimensional Einstein-like manifolds and curvature homogeneity, Geom. Dedicata 54 (1995) 225-243], we improve a result from [P. Bueken, L. Vanhecke, Three- and four-dimensional Einstein-like manifolds and homogeneity, Geom. Dedicata 75 (1999) 123-136] and we announce the final solution of the classification problem for 4-dimensional homogeneous D'Atri spaces.     

The Möbius function of permutations with an indecomposable lower bound

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

On rational isotropic congruences of lines

The aim of this paper is to show a way to find an explicit parametrization of rational isotropic congruences of lines in Euclidean three-space It will be shown that also the focal surfaces of these congruences admit a rational parametrization. Furthermore, the close relation of isotropic congruences of lines to minimal surfaces will be used to find the related polynomial minimal surfaces.     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

Conformal hypersurfaces with the same third fundamental form

We classify all hypersurfaces in a Euclidean space which allow conformal deformations, other than the ones obtained through conformal diffeomorphisms of the Euclidean space, preserving the third fundamental form.     

Conformal vector fields with respect to the Sasaki metric tensor

The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.