  We study geometric and topological properties of the image of a smooth
submanifold of $\mathbb{R}^{n}$ under a bi-Lipschitz map to $\mathbb{R}^{m}$.
In particular, we characterize how the dimension, diameter, volume, and reach
of the embedded manifold relate to the original. Our main result establishes a
lower bound on the reach of the embedded manifold in the case where $m \le n$
and the bi-Lipschitz map is linear. We discuss implications of this work in
signal processing and machine learning, where bi-Lipschitz maps on
low-dimensional manifolds have been constructed using randomized linear
operators.
