Eighteen Essays in Non-Euclidean Geometry

Description

This book consists of a series of self-contained essays in non-Euclidean geometry in a broad sense, including the classical geometries of constant curvature (spherical and hyperbolic), de Sitter, anti-de Sitter, co-Euclidean, co-Minkowski, Hermitian geometries, and some axiomatically defined geometries. Some of these essays deal with very classical questions and others address problems that are at the heart of present day research, but all of them are concerned with fundamental topics. All the essays are self-contained and most of them can be understood by the general educated mathematician. They should be useful to researchers and to students of non-Euclidean geometry, and they are intended to be references for the various topics they present. Keywords: Non-Euclidean geometry, spherical geometry, hyperbolic geometry, Busemann type geometry, curvature, geographical map, non-euclidean area, non-euclidean volume, Brahmagupta’s formula, Ptolemy’s theorem, Casey’s theorem, Sforza’s formula, Seidel’s problem, infinitesimal rigidity, static rigidity, Pogorelov map, Maxwell–Cremona correspondence, exterior hyperbolic geometry, de Sitter geometry, non-Euclidean conics, bifocal properties, focus-directrix properties, pencils of conics, projective geometry, convexity, duality, transition, Hermitian trigonometry, complex projective trigonometry, shape invariant, metric plane projective-metric plane Foreword......Page 6 Introduction......Page 8 Prologue......Page 22 Contents......Page 26 Part I. Spherical and hyperbolic geometries......Page 30 1. Area in non-Euclidean geometry......Page 32 2. The area formula for hyperbolic triangles......Page 56 3. On a problem of Schubert in hyperbolic geometry......Page 76 4. On a theorem of Lambert: medians in spherical and hyperbolic geometries......Page 86 5. Inscribing a triangle in a circle in spherical geometry......Page 96 6. Monotonicity in spherical and hyperbolic triangles......Page 110 7. De Tilly's mechanical view on hyperbolic and spherical geometries......Page 122 8. The Gauss–Bonnet theorem and the geometry of surfaces......Page 142 9. On the non-existence of a perfect map from the 2-sphere to the Euclidean plane......Page 154 10. Area preserving maps from the sphere to the Euclidean plane......Page 164 11. Area and volume in non-Euclidean geometry......Page 180 12. Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry......Page 220 Part II. Projective geometries......Page 264 13. Contributions to non-Euclidean geometry I......Page 266 14. Notes on Eduard Study's paper ``Contributions to non-Euclidean geometry I''......Page 282 15. Spherical and hyperbolic conics......Page 292 16. Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions......Page 350 Part III. Projective geometries......Page 440 17. Hermitian trigonometry......Page 442 18. A theorem on equiareal triangles with a fixed base......Page 456 List of contributors......Page 468