I'm glad I could help! > could you please collect the definitions of (i) Hyperbolic space (ii) Minkowski space (iii) Lorentzian manifold (iv) Pseudo Riemannian

I'm glad I could help! > could you please collect the definitions of (i) Hyperbolic space (ii) Minkowski space (iii) Lorentzian manifold (iv) Pseudo Riemannian manifold? Sure: A Riemannian manifold is a smooth manifold M together with a positive-definite metric g on M. A hyperbolic manifold is a Riemannian manifold M of constant negative sectional curvature. It is often required in addition that M be a complete manifold. A pseudo-Riemannian manifold is a smooth manifold M together with a non-degenerate metric g on M (of arbitrary signature). A Lorentzian manifold of dimension n is a pseudo-Riemannian manifold (of dimension n) whose metric has signature (1, n-1) or, equivalently, (n-1, 1). (n+1)-dimensional Minkowski space is the Lorentzian manifold (ℝ^(n+1), η), where η = - (dx⁰)² + (dx¹)² + … + (dx^n)² with respect to the standard coordinates (x⁰, x¹, …, x^n) on ℝ^(n+1).