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.
