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

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