The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric
Résumé
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 the following: S may be of eight different types, including the generic type S=R.Id, and the Kähler and hyperkähler types where S is respectively isomorphic to the complex field C or to the quaternions H. This is a result on real, semi-simple algebras with involution. Then, 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, and give the constraints imposed to the Ricci curvature when A is not reduced to R.Id.
Domaines
Géométrie différentielle [math.DG]
Origine : Fichiers produits par l'(les) auteur(s)