Geometric properties of conformal transformations on $$\mathbb {R}^{p,q}$$ R p , q

Abstract

We show that conformal transformations on the generalized Minkowski space R p,q map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when p or q is 0, and that this action has exactly three orbits if p, q = 0. Then we extend these results to hyperboloids and affine planes of arbitrary dimension. These properties generalize the well-known properties of Möbius (or fractional linear) transformations on the complex plane C.

A reassessment of the Neotropical genus Pseudonannolene Silvestri, 1895: cladistic analysis, biogeography, and taxonomic review (Spirostreptida: Pseudonannolenidae)

“…Most species of Pseudonannolene are found in outcrops of limestone rocks (Trajano 1987;Trajano & Gnaspini-Netto 1991;Pinto-da-Rocha 1995;Trajano et al 2000;Freitas et al 2004;Gallo & Bichuette 2019). The species P. ambuatinga, P. lundi Iniesta & Ferreira, 2015, and P. spelaea are restricted to caves, presenting troglomorphisms such as depigmentation and body size reduction (Iniesta & Ferreira 2013a, 2015. The troglophilic species P. microzoporus, P. robsoni, and P. tocaiensis have been found near vegetal debris (for instance, rotten trunks and litter) or guano of Desmodus rotundus (Geoffroy, 1810) (Chiroptera) inside caves, while P. robsoni, P. leopoldoi P.…”