On the algebra of parallel endomorphisms of a pseudo-Riemannian metric

Charles Boubel (1)

1 IRMA - Institut de Recherche Mathématique Avancée

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 E, which is also the commutant of the holonomy group of g. As any associative algebra, E is the sum of its radical and of a semi-simple algebra S. This S may be of eight different types, see my article 'The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric: semi-simple part'. 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 S and {N} are included in E is non-empty. We parametrize it. Generically, the holonomy algebra of those metrics is the full commutant $o(g)^{S\cup\{N\}}$ and then, apart from some ``degenerate'' cases, E is the direct sum of S and (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 treats the case where the radical of E is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.

Mots clés: nilomorphic" functions; semi-simple associative algebra; Pseudo-Riemannian; Kähler; hyperkähler; parakähler metrics; holonomy group; parallel endomorphism; "nilomorphic" functions; nilpotent endomorphism; commutant; Ricci curvature; real algebra with involution; semi-simple associative algebra.

Domaines: Géométrie différentielle [math.DG]

Charles Boubel. On the algebra of parallel endomorphisms of a pseudo-Riemannian metric. Journal of Differential Geometry, 2015, 99 (1), pp.77-123. ⟨10.4310/jdg/1418345538⟩. ⟨hal-00721469v7⟩