C¹-isometric embeddings of Riemmanian 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 → W that is long (g≤ f*h) can be C⁰-approximated by a C¹ isometric embedding F: (V,g) → (W,h).