There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: how can one recognize those metric spaces that are isometrically embeddable into Hilbert spaces?

Later edit: I have removed two paragraphs from my original question, which created a lot of confusion among those who answered it. I take responsibility for mixing "metric spaces" isometries and "differential geometry" isometries. I apologize.