$C^{1}$-isometric embeddings of Riemannian spaces in Lorentzian spaces

TL;DR

This paper proves that any spacelike embedding of a compact Riemannian manifold into a Lorentzian manifold can be approximated by a $C^{1}$ isometric embedding, advancing the understanding of embeddings in Lorentzian geometry.

Contribution

It establishes the $C^{0}$-approximation of spacelike embeddings by $C^{1}$ isometric embeddings in Lorentzian spaces, a novel result in differential geometry.

Findings

Any spacelike embedding can be approximated by a $C^{1}$ isometric embedding.

The approximation holds for embeddings of compact Riemannian manifolds into Lorentzian manifolds.

The result applies to long embeddings satisfying $g leq f^{*}h$.

Abstract

For any compact Riemannian manifold $(V,g)$ and any Lorentzian manifold $(W,h)$, we prove that any spacelike embedding $f: V \rightarrow W$ that is long ($g\leq f^{*}h$) can be $C^{0}$-approximated by a $C^{1}$ isometric embedding $F: (V,g) \rightarrow (W,h)$.