Hyperbolic manifold
![A perspective projection of a dodecahedral tessellation in H3. This is an example of what an observer might see inside a hyperbolic 3-manifold.](/Images/godic/202501/19/Hyperbolic_orthogonal_dodecahedral_honeycomb5501.png")
![The Pseudosphere. Each half of this shape is a hyperbolic 2-manifold (i.e. surface) with boundary.](/Images/godic/202501/19/The_Pseudosphere5501.jpg")
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.