A Santalo's formula in L-P
Graciela Silvia · Hokkaido Mathematical Journal · 1989
It is show that a formula by Santal\'o on hyperbolic space of curvature -1 holds for Lorentz-Poincar\^e upper half space with curvature 1. Introduction We call L-P plane, or simply L-P, relating to Lorentz- Poincare, to the upper half space with the metric ds^{2}= \frac{dx^{2}-dyz}{y^{2}} .The curvature of the L-P plane is 1.If z is a complex variable, the group SL(2) acts on the upper half plane {\rm Im}(z)>0 as the transformation group.z'= \frac{az+b}{pz+q} aq-bp=1Where a , b , p , q are real numbers.This is the classical Poincar\^e model for non-euclidean hyperbolic geometry.In the first section we introduce the double numbers, see [1], [7] and [8].The referee observed that the refer- ence [6], pag.166, is appropiate.We show that substitution in the above transformation of the complex variable by a double number variable we obtain the Lorentz-Poincar\'e geometry.We also find relationship between double numbers, curvature and geodesies.Our main results is the integral formula in the third section.Along the second section we obtain different expressions for the density of points, pair of points, geodesies, pair of geodesies, and kinematic density as is customary in integral geometry.Some of them will be used in the following section.