Fubini-Study metrics and Levi-Civita connections on quantum projective spaces

Abstract

We introduce analogues of the Fubini-Study metrics and the corresponding Levi-Civita connections on quantum projective spaces, following the approach of Beggs and Majid. We define the quantum metrics as two-tensors, symmetric in the appropriate sense, in terms of the differential calculi introduced by Heckenberger and Kolb. We define connections on these calculi and show that they are torsion free and cotorsion free, where the latter condition uses the quantum metric and is a weaker notion of metric compatibility.

Levi-Civita connections for conformally deformed metrics on tame differential calculi

“…We refer to [6], [27] and references therein and the book [9] for a comprehensive account. Following this line, existence of a Levi-Civita connection on any quantum projective space ( for the Fubini-Study metric ) has been proved in [28]. For q-deformed connections on S 3 q , we refer to the work of Landi, Arnlind and Ilwale ( [4] ) while for Levi-Civita connections on finite metric spaces and graphs, we refer to Chapter 8 of [9] and the paper [14] by Sitarz et al Finally, yet another approach to study Levi-Civita connections on quasi-commutative algebras has been initiated in [33] and [5].…”