[Paper Review] Basic coordinate-free non-Euclidean geometry

Sasha Anan’in, Carlos H. Grossi|arXiv (Cornell University)|Jul 2, 2011

Computational Geometry and Mesh Generation3 citations

TL;DR

This paper presents a coordinate-free approach to non-Euclidean geometry, emphasizing intrinsic geometric structures over coordinate systems. It develops foundational tools in differential geometry, Riemannian geometry, and complex hyperbolic geometry using sheaves, manifolds, and tangent bundles, culminating in a coordinate-free treatment of geodesics, metrics, and curvature on spheres and hyperbolic spaces.

ABSTRACT

These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.

Motivation & Objective

• To develop a geometric framework independent of arbitrary coordinate choices, motivated by Hermann Weyl's critique of coordinates as 'an act of violence'.
• To reformulate classical non-Euclidean geometries—spherical and hyperbolic—using intrinsic differential geometric tools.
• To provide a coordinate-free foundation for Riemannian geometry, focusing on tangent bundles, metrics, and geodesics.
• To explore the geometry of the Riemann sphere, Grassmannians, and complex hyperbolic spaces through intrinsic structures.
• To reframe historical developments in non-Euclidean geometry, particularly Bolyai and Lobachevsky’s work, within a modern, coordinate-free context.

Proposed method

• Uses sheaves of smooth functions and $C^inity$-manifolds to define geometric objects without reliance on coordinates.
• Applies the tangent bundle construction and differential maps to define tangent vectors and geodesics intrinsically.
• Employs Taylor sheaves and prevarieties to formalize local geometric behavior and smooth structures.
• Utilizes the Riemann sphere and stereographic projection to model spherical geometry without embedding in Euclidean space.
• Applies Grassmannian constructions to describe spaces of subspaces and geometric configurations.
• Introduces the concept of 'geometry on the absolute' to unify spherical and hyperbolic structures via duality and metric invariance.

Experimental results

Research questions

• RQ1 How can non-Euclidean geometries be formulated without relying on coordinate systems?
• RQ2 What intrinsic geometric structures underlie the spherical and hyperbolic planes, and how can they be unified?
• RQ3 How do geodesics and metrics emerge naturally from tangent bundle and differential structures?
• RQ4 What is the role of duality and the absolute in unifying spherical and hyperbolic geometry?
• RQ5 How can historical developments in non-Euclidean geometry be reinterpreted through a modern, coordinate-free lens?

Key findings

• The paper establishes that non-Euclidean geometries, including spherical and hyperbolic, can be rigorously formulated using sheaves and $C^inity$-manifolds without coordinate dependence.
• It demonstrates that geodesics on the sphere and hyperbolic plane arise naturally from the differential structure of the tangent bundle and metric compatibility.
• The Riemann sphere is shown to provide a natural compactification of the complex plane, enabling a unified treatment of spherical and complex hyperbolic geometry.
• The concept of 'geometry on the absolute' reveals a deep duality between spherical and hyperbolic geometries via metric invariance and projective structure.
• The paper reinterprets the historical discovery of non-Euclidean geometry by Bolyai and Lobachevsky as a coordinate-free insight into curvature and parallel lines.
• It confirms that the choice of coordinates obscures geometric essence, and that intrinsic methods reveal symmetries and conservation laws more transparently.