论文标题
$ 3 $ -Dimensional Riemannian submanifolds的内在特征的$ \ Mathbb {r}^4 $
Intrinsic Characterization of $3$-dimensional Riemannian submanifolds of $\mathbb{R}^4$
论文作者
论文摘要
众所周知，当且仅当存在满足高斯和codazzi方程的对称的2张量字段时，就可以将$ M $维的Riemannian歧管当地嵌入$ m+1 $二维的欧几里得空间。在本文中，我们证明了魏斯，托马斯和里维茨先前获得的两个已知内在条件足以确保在$ m = 3 $的某些通用条件下在某些通用条件下存在这种对称的2张量。请注意，在$ M = 3 $的情况下，满足高斯方程的对称2 tensor场并不能自动满足Codazzi方程，这与$ M \ geq 4 $不同。在我们的证明中，符号方法是一种在古典不变理论中众所周知的著名工具，起着重要作用。作为我们结果的应用，我们考虑了$ 3 $维扭曲的产品Riemannian歧管是否可以将它们局部嵌入到$ \ mathbb {r}^4 $中。在某些情况下，Monge-ampère方程自然会出现。
It is well known that an $m$-dimensional Riemannian manifold can be locally isometrically embedded into the $m+1$-dimensional Euclidean space if and only if there exists a symmetric 2-tensor field satisfying the Gauss and Codazzi equations. In this paper, we prove that two known intrinsic conditions, which were obtained previously by Weiss, Thomas and Rivertz, are sufficient to ensure the existence of such symmetric 2-tensor field under certain generic condition when $m=3$. Note that, in the case $m=3$, a symmetric 2-tensor field satisfying the Gauss equation does not satisfy the Codazzi equation automatically, which is different from the cases $m \geq 4$. In our proof, the symbolic method, which is a famous tool known in classical invariant theory, plays an important role. As applications of our result, we consider $3$-dimensional warped product Riemannian manifolds whether they can be locally isometrically embedded into $\mathbb{R}^4$. In some case, the Monge-Ampère equation naturally appears.
