  We show that compact Riemannian manifolds, regarded as metric spaces with
their global geodesic distance, cannot contain a number of rigid structures
such as (a) arbitrarily large regular simplices or (b) arbitrarily long
sequences of points equidistant from pairs of points preceding them in the
sequence. All of this provides evidence that Riemannian metric spaces admit
what we term loose embeddings into finite-dimensional Euclidean spaces:
continuous maps that preserve both equality as well as inequality.
  We also prove a local-to-global principle for Riemannian-metric-space loose
embeddability: if every finite subspace thereof is loosely embeddable into a
common $\mathbb{R}^N$, then the metric space as a whole is loosely embeddable
into $\mathbb{R}^N$ in a weakened sense.
