[Paper Review] A Remark on the Projective Geometry of Constant Curvature Spaces

Athanase Papadopoulos, Sumio Yamada|arXiv (Cornell University)|Sep 17, 2012

Advanced Differential Geometry Research5 references3 citations

TL;DR

This paper establishes a unified projective geometric framework for constant curvature spaces—Euclidean, spherical, and hyperbolic—by embedding them into Minkowski space ℝⁿ,¹ and defining geometry-specific cross ratios. It proves that projective maps from the upper hemisphere (spherical) and hyperbolic space to the Euclidean plane preserve cross ratios, leading to new generalized Beltrami-Klein models that realize hyperbolic space as isometric to geodesic balls in spherical and hyperbolic geometry with Hilbert metrics.

ABSTRACT

We highlight the relation between the projective geometries of $n$-dimensional Euclidean, spherical and hyperbolic spaces through the projective models of these spaces in the $n+1$-dimensional Minkowski space, using a cross ratio notion which is proper to each of the three geometries.

Motivation & Objective

• To unify the projective geometry of constant curvature spaces—Euclidean, spherical, and hyperbolic—through a common framework in Minkowski space.
• To define geometry-specific cross ratios for each space, preserving invariance under projective transformations.
• To establish that projective maps from spherical and hyperbolic spaces to the Euclidean plane preserve cross ratios, thus acting as perspectivities.
• To introduce generalized Beltrami-Klein models for hyperbolic space using Hilbert metrics on geodesic balls in spherical and hyperbolic geometry.
• To demonstrate isometry between hyperbolic space and geodesic balls equipped with spherical or hyperbolic Hilbert metrics.

Proposed method

• Define three distinct cross ratios: Euclidean (ratio of distances), spherical (ratio of sines of distances), and hyperbolic (ratio of hyperbolic sines of distances).
• Use the projective model of the sphere as the upper hemisphere in ℝⁿ,¹, with radial projection to the plane {xₙ₊₁ = 1}.
• Use the projective model of hyperbolic space as the upper sheet of the unit hyperboloid in Minkowski space ℝⁿ,¹, with radial projection to {xₙ₊₁ = 1}.
• Prove that the radial projection maps preserve cross ratios by relating spherical/hyperbolic distances to Euclidean distances in the plane via trigonometric and hyperbolic identities.
• Define spherical and hyperbolic Hilbert metrics on geodesic balls as the logarithm of the respective cross ratios of quadruples (x, y, b(x,y), b(y,x)).
• Establish that the projective maps are isometries between the Hilbert metric on geodesic balls and the standard hyperbolic metric.

Experimental results

Research questions

• RQ1 How can the projective geometry of Euclidean, spherical, and hyperbolic spaces be unified via a common model in Minkowski space?
• RQ2 What is the appropriate definition of cross ratio in each of the three constant curvature geometries, and how does it behave under projective transformations?
• RQ3 Can radial projections from the upper hemisphere and hyperbolic space to the Euclidean plane be shown to preserve cross ratios, thus acting as perspectivities?
• RQ4 Do the spherical and hyperbolic Hilbert metrics on geodesic balls yield models isometric to hyperbolic space?
• RQ5 Can generalized Beltrami-Klein models be constructed for hyperbolic space using these Hilbert metrics on curved geodesic balls?

Key findings

• The radial projection from the upper hemisphere of Sⁿ to the plane {xₙ₊₁ = 1} preserves the spherical cross ratio, making it a perspectivity.
• The radial projection from the hyperbolic space ℍⁿ to the same plane preserves the hyperbolic cross ratio, establishing it as a perspectivity.
• The spherical Hilbert metric Hₛᵨ on a geodesic ball Bₛᵨ of radius ρ ≤ π/2 in Sⁿ is isometric to hyperbolic space ℍⁿ.
• The hyperbolic Hilbert metric Hₕᵨ on a geodesic ball Bₕᵨ in ℍⁿ is isometric to hyperbolic space ℍⁿ, confirming the generalized Beltrami-Klein model.
• The projective maps Pₛ and Pₕ are isometries between the Hilbert metrics on geodesic balls and the standard hyperbolic metric on ℍⁿ.
• The construction provides a new class of generalized Beltrami-Klein models for hyperbolic space, based on spherical and hyperbolic Hilbert metrics on geodesic balls.