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 {\em i.e.\@} sections of $\End(TM)$ may be parallel for $g$. They form an associative algebra $\goth e$, which is also the commutant of the holonomy group of $g$. We study it. As any associative algebra, $\goth e$ is the sum of its radical and of a semi-simple algebra $\goth s$. We show the following: $\goth s$ may be of eight different types, including the generic type $\goth s={\mathbb R}\Id$, and the Kähler and hyperkähler types $\goth s\simeq{\mathbb C}$ and $\goth s\simeq{\mathbb H}$. For $N$ any self adjoint nilpotent element of the commutant of such an $\goth s$ in $\End(TM)$, the set of germs of metrics the holonomy group of which is included in the commutant of $\goth{s}\cup\{N\}$ in O$^0(g)$ is non empty. We parametrise it. Generically, the holonomy group of those metrics is the full commutant O$^0(g)^{\goth{s}\cup\{N\}}$. Apart from some ''degenerate'' cases, the algebra of the parallel endomorphisms of those metrics is $\goth s \times \langle N{\mathbb R}angle$. To prove it, we introduce an analogy with complex Differential Calculus, the ring ${\mathbb R}[X]/(X^n)$ replacing the field ${\mathbb C}$. This describes totally the local situation when the radical of $\goth e$ 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 $\goth e$ is non trivial.
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