Remark on holonomy groups of pseudo-Riemannian manifolds of signature (2,n+2)

Details Remark on holonomy groups of pseudo-Riemannian manifolds of signature (2,n+2) Remark on holonomy groups of pseudo-Riemannian manifolds of signature (2,n+2)

2004-06-21

arxiv.org/abs/math/0406397v2

Mathematics

Differential Geometry

8 pages

Scientific paper

We consider one type of weakly-irreducible not irreducible subalgebras of $\so(2,n+2)$. Each Lie algebra $\g^\h$ of this type is uniquely defined by the associated subalgebra $\h\subset\so(n)$. For any $\h\subset\so(n)$ we realize $\g^\h$ as the holonomy algebra of a pseudo-Riemannian manifold of signature $(2,n+2)$. This shows the principal difference from the case of Lorentzian manifolds, where the analogous subalgebra $\h\subset\so(n)$ associated to the holonomy algebra has to be the holonomy algebra of a Riemannian manifold.

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