Complete spacelike submanifolds in a locally symmetric semi-Riemannian space

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold [Formula: see text] of [Formula: see text].

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

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.

CHARACTERIZATIONS OF LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN EINSTEIN SPACETIMES

Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

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.

Isoparametric hypersurfaces with four principal curvatures, II

In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) inS15is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively asof OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraintm2≥2m1-1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

Isoparametric hypersurfaces with four principal curvatures, II

In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint m2≥ 2m1 -1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

Isoparametric Hypersurfaces with four Principal Curvatures Revisited

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi’s expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

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.