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Thank you, It does appear I am hopefully confused between hyperbolic and Lorentzian :) Thanks for patiently showing me where I am going wrong. For my selfish be
by bollu 6y ago
Thank you, It does appear I am hopefully confused between hyperbolic and Lorentzian :) Thanks for patiently showing me where I am going wrong. For my selfish benefit, could you please collect the definitions of (i) Hyperbolic space (ii) Minkowski space (iii) Lorentzian manifold (iv) Pseudo Riemannian manifold?
- codethief 6y agoI'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).
- bollu 6y agoAnd what is the precise relationship between hyperbolic space and Minkowski space? am I right in understanding that one can _embed_ a model of hyperbolic space (the hyperboloid moidel, obeying the equation `-x0^2 + x1^2 + ... xn^2 = 0` as a subspace of the Minkowski space? But Minkowski space is far larger than hyperbolic space?
- codethief 6y agopa7x1 already answered that question: https://news.ycombinator.com/item?id=23549361 https://news.ycombinator.com/item?id=23549361