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

TL;DR

This paper establishes intrinsic conditions that guarantee the local isometric embedding of 3-dimensional Riemannian manifolds into 4-dimensional Euclidean space, highlighting differences from higher dimensions and using classical invariant theory tools.

Contribution

It proves that two known intrinsic conditions are sufficient for embedding 3D manifolds into R^4 under certain generic assumptions, addressing the unique case where the Codazzi equation is not automatically satisfied.

Findings

Intrinsic conditions ensure embedding in R^4 for 3D manifolds.

The symbolic method is crucial in the proof.

Applications include embedding of warped product manifolds and connections to Monge-Ampère equations.

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.