Is there any ~~example of a~~ compact manifold $M$ of dimension $n>10000$
such that

$M$ admits an embedding into $\mathbb R^{n+2}$,

$M$ is hyperbolic; i.e., it admits a Riemannian metric with curvature $\equiv -1$?

More generally, are there some criteria sufficient for existence of embedding of compact manifolds $M^n$ of large dimension in $\mathbb R^{n+2}$?