On submanifolds with parallel Möbius second fundamental form in the unit sphere

In this paper, we study umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form. As one of our main results, we establish a complete classification for such submanifolds under the additional condition of codimension two.

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.

Classification of Möbius Isoparametric Hypersurfaces in 4

Let Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.