A three dimensional Lorentzian hypersurface x : M_1~3→ R_1~4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of R_1~4. Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.
从根分布的角度,对齐性二维球面分类结果给出比Bando和Ohnita(J Math Soc Japan,1987,39:477)更加明显的刻画,求出决定齐性二维球面的SU(2)轨道的李群多项式表示的显式表达式,证明复射影空间中SU(2)轨道的维数取决于一个对应的扩大复平面系数上的一元n次方程的重根和负共轭倒数根对分布,把SU(2)轨道维数归结为黎曼球面上n个点是否重合或成为对径点的问题.也初步研究了SU(2)三维轨道性质与根分布的关系.
Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This property is conformal invariant;hence we study them in the framework of Mbius geometry,and restrict to three-dimensional Wintgen ideal submanifolds in S5.In particular,we give Mbius characterizations for minimal ones among them,which are also known as(3-dimensional)austere submanifolds(in 5-dimensional space forms).
In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper"Homogeneous harmonic maps into complex projective spaces.Tokyo J Math,1990,13:87–116".
Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v(k ≥ 1) of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.
