论文标题
高斯地图的曲率，用于空间形式的正常平面子曼叶
Curvature of the Gauss map for normally flat submanifolds in space forms
论文作者
论文摘要
对于具有空间形式平坦的正常束的子手机，有一个正常基础的基础，同时将相应的Weingarten运算符对角线，并且这些操作员满足简单的Codazzi对称性。当第二种基本形式的相对无效指数为零时，由Obata定义的高斯图的图像是通用的Grassmannian歧管的子手，其诱导的度量由第三个基本形式给出，对应于这些算子的平方之和。在这种情况下，我们表明，高斯图像的Riemann曲率张量完全由原始Submanifold的曲率和Weingarten算子确定，其形式类似于与原始emperementry the Oriptic egegium相关的理论类似物，该几何形状与原始empedding的外部几何相关。为此，我们得出了诱导的Levi-Civita连接的差异向量，并用广义的Kulkarni-Nomizu产品表达Riemann曲率张量。
For a submanifold with flat normal bundle in a space form there is a normal orthonormal basis that simultaneously diagonalizes the corresponding Weingarten operators, and at which these operators satisfy a simple Codazzi symmetry. When the second fundamental form has zero index of relative nullity, the image of the Gauss map as defined by Obata is a submanifold of a generalized Grassmannian manifold, with induced metric given by the third fundamental form, corresponding to the sum of the squares of these operators. In this case, we show that the Riemann curvature tensor of the Gauss image is completely determined by the curvature and Weingarten operators of the original submanifold, in a form analogous to a theorema egregium that relates the intrinsic geometry of the Gauss map with the extrinsic geometry of the original embedding. For this, we derive the difference vector of the induced Levi-Civita connections and express the Riemann curvature tensor with a generalized Kulkarni-Nomizu product.
