Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 052, 35 pages      arXiv:2210.02251      https://doi.org/10.3842/SIGMA.2023.052

Single-Valued Killing Fields of a Meromorphic Affine Connection and Classification

Alexis Garcia
Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, France

Received November 09, 2022, in final form July 17, 2023; Published online July 27, 2023

Abstract
We give a geometric condition on a meromorphic affine connection for its Killing vector fields to be single valued. More precisely, this condition relies on the pole of the connection and its geodesics, and defines a subcategory. To this end, we use the equivalence between these objects and meromorphic affine Cartan geometries. The proof of the previous result is then a consequence of a more general result linking the distinguished curves of meromorphic Cartan geometries, their poles and their infinitesimal automorphisms, which is the main purpose of the paper. This enables to extend the classification result from [Biswas I., Dumitrescu S., McKay B., Épijournal Géom. Algébrique 3 (2019), 19, 10 pages, arXiv:1804.08949] to the subcategory of meromorphic affine connection described before.

Key words: meromorphic affine connections; Killing vector fields; infinitesimal automorphisms; Cartan geometries.

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