论文标题
关于无效的子曼叶的平均曲率
On the mean curvature of submanifolds with nullity
论文作者
论文摘要
在本文中，我们研究了与圆柱体相对无效指数的等距浸入的几何条件。在某些类别的椭圆表面上，有大量的非圆柱形$ n $二维最小的子手机，带有相对无效$ n-2 $的索引，由Dajczer和Florit \ cite \ cite {df2}进行了充分描述。与此相反，我们证明了任何编纂的空间形式的非微小$ n $维度的次符号是本地圆柱体，前提是它们具有秩$ n-2 \ geq2的完全测量分布，$ n-2 \ geq2（$ cyq2）$ nullity分布中包含的平均曲率矢量场的长度是沿每个平均值沿每个叶片的长度。尺寸$ n = 3 $的情况很特别。我们表明，在满足上述特性的球体中存在椭圆形的三维亚策略。实际上，我们提供了三维亚策略的参数化，作为欧几里得空间中最小表面的单位切线束，其第一个曲率椭圆椭圆形无处不在一个圆，第二个圆圈无处不在。此外，我们向submanifolds提供了几种应用，其平均曲率矢量场的长度恒定，条件弱于平行。
In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fully described by Dajczer and Florit \cite{DF2} in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal $n$-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank $n-2\geq2,$ which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension $n=3$ turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.
