The algebra of the parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part
arXiv:1402.6642 · doi:10.1017/S0305004115000304
Abstract
On a (pseudo-)Riemannian manifold (MM,g), some fields of endomorphisms i.e. sections of End(TMM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. Here we study S: it may be of eight different types, including the generic type S=R.Id, and the Kähler and hyperkähler types 'S isomorphic to C' and 'S isomorphic to the quaternions'. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrise it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields
19 pages. This is a part of the preprint arXiv:1207.6544 "The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric", which we needed to put online separately. The writing is a little improved with respect to the previous version. This article will appear in 'Mathematical proceedings of the Cambridge Philosophical Society'