From the Nash Embedding theorem e.g. see
https://en.wikipedia.org/wiki/Nash_embedding_theorem
an $m$ dimensional Riemannian manifold has an isometric embedding into a dimension of $\frac{m(3m+11)}{2}$ or $\frac{m(m+1)(3m+11)}{2}$ depending on if its compact or not.
My questions are:
- does there always exist an isometric embedding, in the sense of the of the Nash theorem above, of an $m$ dimensional Riemannian manifold into $\frac{m(m+1)}{2}$ space?
- 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 $\frac{m(m+1)}{2}$ space for certain $m$.