From the Nash Embedding theorem e.g. see
an m dimensional Riemannian manifold has an isometric embedding into a dimension of m(3m+11)/2 or m(m+1)(3m+11)/2 depending on if its compact or not.
My questions are: 1) does there always exist an isometric embedding, in the sense of the of the Nash theorem above, of an m dimensional Riemannian manifold into m(m+1)/2 space? 2) Is the first question known to be false for any m?
As far as I'm aware the answer to the first question is probably true because there are specific known examples of m manifolds into m(m+1)/2 space for certain m.