Conformal Poincar\'e interpretation of a~degenerate geometry in a~dual variable plane, and Norden's normalization theory

S. Yu. Petropavlovskaya

Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 85-98

Résumé

In this paper we consider partial cases of autopolar normalization of the quadrics $-(x^1)^2+(x^3)^2-(x^4)^2=0$ and $(x^1)^2+(x^3)^2-(x^4)^2=0$ in three-dimensional projective space. We obtain the conformal Poincaré interpretations in the dual plane (the absolute is formed by a pair of mutually intersected straight lines and a pair of mutually intersected imaginary straight lines). We study connections which arise from these autopolar normalizations and prove that their parallel translations satisfy the translation law of dual variable. We find basic operators of infinitesimal transformations which generate a 7-dimensional transformation group $\mathcal G_7$. The group $\mathcal G_7$ contains a 6-dimensional subgroup of homographic mappings of dual variable with dual coefficients. We study differentialgeometric properties of these operators.