All closed 2D manifolds except the sphere, torus, klein bottle, and projective plane are hyperbolic. Thus, e.g., the double torus is a hyperbolic manifold. However, when embedding a double torus in Euclidean space, it has obvious regions of positive curvature (just as an embedding regular torus has regions of positive curvature despite being a Euclidean manifold).
Meanwhile, if you embed a sphere in hyperbolic 3-space, it retains its inherent positive curvature.
So, is there a way to embed a double torus (or some other closed hyperbolic surface) into $H^3$ such that it has negative curvature everywhere?
If not, is there any other way to embed a closed hyperbolic $n$-manifold in a space of $n+1$ dimensions such that it retains negative curvature everywhere?