设 x:M→R^(n+1)是凸域ΩR^n 上的严格凸函数 x_(n+1)=f(x_1,…,x_n)定义的一个局部强凸超曲面.如果 f 是下面方程的解,则称 M 为α相对极值超曲面:△ρ=(2-nα)/2(‖▽ρ‖~2)/ρ,ρ:=(det((a^2f)/(ax_iax_j)))^(1/(n+2)).2007年,贾和李证明了存在一个仅依赖于维数 n 的正常数 K(n),如果|α|≥K(n),那么欧氏完备的α相对极值超曲面是椭圆抛物面.本文中我们利用 Calabi 度量给出了这个定理的一个简单证明.
In this paper we study a class of metrics with some compatible almost complex structures on the tangent bundle TM of a Riemannian manifold (M,g), which are parallel to those in [10]. These metrics generalize the classical Sasaki metric and Cheeger-Gromoll metric. We prove that the tangent bundle TM endowed with each pair of the above metrics and the corresponding almost complex structures is a locally conformal almost K¨ahler manifold. We also find that, when restricted to the unit tangent sphere bundle, these metrics and corresponding almost complex structures define new examples of contact metric structures.
Given an immersed submanifold x ： M^M → S^n in the unit sphere S^n without umbilics, a Blaschke eigenvalue of x is by definition an eigenvalue of the Blaschke tensor of x. x is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the MSbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the Blaschke isoparametric hypersurfaces in S^5 with more than two distinct Blaschke eigenvalues are necessarily Mobius isoparametric.
