Trigonometric laws on Lorentzian sphere \(S^ 2_ 1\) (Q2266456)

The Lorentzian space \({\mathcal L}_ 3\) is the vector space with scalar product \(x_ 1y_ 1+x_ 2y_ 2-x_ 3y_ 3\). The author defines the notions of vector product, distance, arc-length, area, angle (a complex number) and sphere (it is a hyperboloid in the corresponding Euclidean space) in this context. The geometry on a sphere is studied: geodesics (they are all closed curves), geodesic triangles and their area. So trigonometry on a Lorentzian sphere is developed. There exists a ''parallel angle formula'' which, however, contrasts with that in hyperbolic geometry.