Define H R ⊂ M to be the upper sheet (ct > 0) of the hyperboloid H R 1 ( n ) = { ( c t , x 1 , … … , x n ) ∈ ∈ M n : c 2 t 2 − − ( x 1 ) 2 − − ⋯ ⋯ − − ( x n ) 2 = R 2 , c t > 0 } in generalized Minkowski space M of spacetime dimension n + 1. This is one of the surfaces of transitivity of the generalized Lorentz group. The induced metric on this submanifold, h R 1 ( n ) = ι ι ∗ ∗ η η , the pullback of the Minkowski metric η under inclusion, is a Riemannian metric. With this metric H R is a Riemannian manifold. It is one of the model spaces of Riemannian geometry, the hyperboloid model of hyperbolic space. It is a space of constant negative curvature −1/ R. The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the n for its dimension. A 2(2) corresponds to the Poincaré disk model, while 3(n) corresponds to the Poincaré half-space model of dimension n.