Stiff connections on pseudo-Euclidean spaces

Abstract

Unless it is a flat connection, an affine connection cannot be at the same time projectively flat (geodesics being straight lines) and conformal (conformal structure being preserved by parallel transport). This impossibility is examplified by the two standard models of hyperbolic geometry: Beltrami and Poincaré models. We define stiff connections by weakening the conformality hypothesis to the requirement that the first order infinitesimal holonomies are infinitesimal isometries. In a given pseudo-Euclidean space, stiff connections are characterized by the choice of a potential and form a continuous family of non-flat connections with surprising properties. In particular, we prove the existence of a unique affine connection on the disk that is geodesically complete, infinitesimally conformal and projectively flat. This uniquely characterized connection achieves a compromise between properties of Beltrami and Poincaré models of the disk.

Speaker Intro

Guillaume Tahar is a BIMSA assistant research fellow. Before joining BIMSA he held a senior postdoctoral fellowship in Weizmann Institute of Science. He contributed to the study of moduli spaces of various flavours of geometric structures on surfaces: translation and dilation structures, flat metrics and cone spherical metrics. The recent research interests of Guillaume Tahar involve linear differential operators, isoresidual fibrations and simplicial arrangements of lines.