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 $a$)