Complete hypersurfaces with constant Möbius scalar curvature

Let [Formula: see text] be an umbilical free hypersurface in the unit sphere [Formula: see text]. Four basic invariants of [Formula: see text], under the Möbius transformation group of [Formula: see text] are the Möbius metric [Formula: see text], the Möbius second fundamental form [Formula: see text], the Blaschke tensor [Formula: see text] and the Möbius form [Formula: see text]. In this paper, we study complete hypersurfaces with constant normalized Möbius scalar curvature [Formula: see text] and vanishing Möbius form [Formula: see text]. By computing the Laplacian of the funtion [Formula: see text], where the trace-free Blaschke tensor [Formula: see text], and applying the well known generalized maximum principle of Omori–Yau, we obtain the following result: [Formula: see text] must be either Möbius equivalent to a minimal hypersurface with constant Möbius scalar curvature, when [Formula: see text]; [Formula: see text] in [Formula: see text], when [Formula: see text]; the pre-image of the stereographic projection [Formula: see text] of the circular cylinder [Formula: see text] in [Formula: see text], when [Formula: see text]; or the pre-image of the projection [Formula: see text] of the hypersurface [Formula: see text] in [Formula: see text], when [Formula: see text].

Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we study regular immersed hypersurfaces in Lorentzian space forms with a conformal metric, a conformal second fundamental form, the conformal Blaschke tensor and a conformal form, which are invariants under the conformal transformation group. We classify all the immersed hypersurfaces in Lorentzian space forms with two distinct constant Blaschke eigenvalues and vanishing conformal form.

A note on Blaschke isoparametric hypersurfaces

In this paper, we prove that a Möbius isoparametric hypersurface is a Blaschke isoparametric hypersurface, and a Blaschke isoparametric hypersurface is a Möbius isoparametric hypersurface provided that the Blaschke tensor has more than two distinct eigenvalues.

Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues inSn+1(1)

Letx:M↦Sn+1(1)be ann (n≥3)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form in a unit sphereSn+1(1), and letAandBbe the Blaschke tensor and the Möbius second fundamental form ofx, respectively. We define a symmetric(0,2)tensorD=A+λBwhich is called the para-Blaschke tensor ofx, whereλis a constant. An eigenvalue of the para-Blaschke tensor is calleda para-Blaschke eigenvalueofx. The aim of this paper is to classify the oriented hypersurfaces inSn+1(1)with two distinct para-Blaschke eigenvalues under some rigidity conditions.

PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

Let A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

A CLASSIFICATION OF HYPERSURFACES WITH PARALLEL PARA-BLASCHKE TENSOR IN Sm+1

In this paper, we classify all immersed hypersurfaces in the unit sphere Sm+1 with parallel para-Blaschke tensor.