# Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?

QUESTION: Let $$n$$ be a natural number. Is it true that there exist $$N(n), D(n) > 0$$ such that any complete $$n$$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $$N$$-dimensional Euclidean space with bi-Lipschitz distortion less then $$D$$?

It is known that an analogous statement for nonnegative sectional curvature is true. See Dose closed Alexandrov space admit a bi-Lipschitz embedding into R^N?

It is also known that non-negative Ricci curvature implies doubling property. Thus, by Assouad's embedding Theorem we have that for every $$0 < a < 1$$ and every natural $$n$$ there exist $$N(n,a), D(n,a) > 0$$ with the following property. For every complete $$n$$-dimensional manifold $$(M,d)$$ of nonnegative Ricci curvature its $$a$$-snowflake $$(M,d^a)$$ can be embedded into $$N$$-dimensional Euclidean space with bi-Lipschitz distortion less then $$D$$. (Here, $$(M,d^a)$$ is the metric with all distances raised to the power $$a$$)