The algebra of parallel endomorphisms of a germ of pseudo-Riemannian metric

Abstract

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) 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. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the K\"ahler and hyperk\"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of such an S in End(TM), the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\cup\{N\}$ in O(g). Apart from some "degenerate" cases, the algebra A is then $S \oplus (N)$, where (N) is the ideal spanned by N. To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X^n) replacing the field C. This describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.