H2×R

H2×R denotes the Riemannian 3-manifold obtained as the Cartesian product H^2 × R, where H^2 is the hyperbolic plane with its standard metric of constant curvature -1. The product metric ds^2 = ds_H^2 + dt^2 makes it a simply connected, complete, homogeneous 3-dimensional Riemannian manifold. As a Thurston geometry, H^2 × R is one of the eight model geometries used to describe certain 3-manifolds.

Its curvature is not constant: planes tangent to the H^2 factor have sectional curvature -1, while planes that involve the R-direction (or mix the two factors) have curvature 0. Thus the geometry is negatively curved in one direction and flat in the others, with sectional curvature values in {-1, 0}. The isometry group is the product Isom(H^2) × Isom(R), reflecting the product structure.

H^2 × R arises as a geometric model for several 3-manifolds. For example, if Σ_g is a closed hyperbolic surface (genus g ≥ 2), then Σ_g × S^1 carries a locally H^2 × R geometry, and many Seifert fibered spaces with a hyperbolic base orbifold admit this geometry as well. More generally, quotients of H^2 × R by discrete groups of isometries yield a wide class of locally H^2 × R manifolds.

In the context of Thurston’s geometrization program, H^2 × R serves as a canonical example of a homogeneous but non-constant-curvature geometry, illustrating how a product structure influences global and geometric properties distinct from constant-curvature models.