Is the Poincaré disk, a compact hyperbolic 2-d manifold. A space with negative curvature and maximal symmetry, the equivalent of a sphere but with negative curv

Is the Poincaré disk, a compact hyperbolic 2-d manifold. A space with negative curvature and maximal symmetry, the equivalent of a sphere but with negative curvature if you will. You may have seen something similar in holography and AdS/CFT context. Or in an Escher painting.