Intrinsic Characterization of $3$-dimensional Riemannian submanifolds of $\mathbb{R}^4$

Abstract: 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-Amp`{e}re equation naturally appears.