1999/10/26 by Francisco J. Herranz, Ramon Ortega, Ramón Ortega +1
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Black Holes and Theoretical Physics #Constant curvature #Curvature #Differentiation of trigonometric functions #Geometry #Mathematical analysis #Mathematics #Noncommutative and Quantum Gravity Theories #Signature (topology) #Spherical trigonometry #Trigonometry #gr-qc #math-ph #math.MG #math.MP
paper · pdf · doi:10.1088/0305-4470/33/24/309
published as J. Phys. A 33 (2000) 4525-4551 (reduced version) · 51 pages, LaTeX
arxiv created 1999/10/26 · openalex publication_date 2000/06/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent trigonometry' and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation . This brings to its logical end the idea of an `absolute trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.