## Journal of Differential Geometry

- J. Differential Geom.
- Volume 99, Number 1 (2015), 77-123.

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

#### Abstract

On a (pseudo-)Riemannian manifold $(\mathcal{M}, g)$, some fields of endomorphisms, i.e. sections of $\mathrm{End}(T \mathcal{M})$, 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 $\mathfrak{s}$. This $\mathfrak{s}$ may be of eight different types;
see C. Boubel, *The algebra of the parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part*. Then,
for any self adjoint nilpotent element $N$ of the commutant of such an $\mathfrak{s}$ in $\mathrm{End}(T \mathcal{M})$, the set of germs of metrics such that $\mathfrak{e} \supset \mathfrak{s} \cup \{ N \}$ is non-empty.
We parametrize it. Generically, the holonomy algebra of those metrics is the full commutant $\mathfrak{o}(g)^{\mathfrak{s} \cup \{ N \} }$ and then, apart from some “degenerate” cases, $\mathfrak{e} = \mathfrak{s} \oplus (N)$,
where $(N)$ is the ideal spanned by $N$. To prove it, we introduce an analogy with complex differential calculus, the ring $\mathbb{R}[X]/(X^n)$ replacing the field $\mathbb{C}$. This treats the case where the radical
of $\mathfrak{e}$ is principal and consists of self adjoint elements. We add a glimpse in the case where this radical is not principal.

#### Article information

**Source**

J. Differential Geom., Volume 99, Number 1 (2015), 77-123.

**Dates**

First available in Project Euclid: 12 December 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1418345538

**Digital Object Identifier**

doi:10.4310/jdg/1418345538

**Mathematical Reviews number (MathSciNet)**

MR3299823

**Zentralblatt MATH identifier**

1321.53023

#### Citation

Boubel, Charles. On the algebra of parallel endomorphisms of a pseudo-Riemannian metric. J. Differential Geom. 99 (2015), no. 1, 77--123. doi:10.4310/jdg/1418345538. https://projecteuclid.org/euclid.jdg/1418345538