The algebra of parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part

Abstract

AbstractOn a (pseudo-)Riemannian manifold (${\mathcal M}$, g), some fields of endomorphisms i.e. sections of End(T${\mathcal M}$) may be parallel for g. They form an associative algebra $\mathfrak e$, which is also the commutant of the holonomy group of g. As any associative algebra, $\mathfrak e$ is the sum of its radical and of a semi-simple algebra $\mathfrak s$. Here we study $\mathfrak s$: it may be of eight different types, including the generic type $\mathfrak s$ = ${\mathbb R}$ Id, and the Kähler and hyperkähler types $\mathfrak s$ ≃ ${\mathbb C}$ and $\mathfrak s$ ≃ ${\mathbb H}$. 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 parametrize it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.