This article is cited in 3 papers

Note on the Holonomy Groups of Pseudo-Riemannian Manifolds

A. S. Galaev

Masaryk University

Abstract: For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$ a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to $2$ and the holonomy algebras of Riemannian and Lorentzian manifolds.

Keywords: holonomy algebra, pseudo-Riemannian manifolds, linear connection, Levi-Cività connection, curvature tensor, Lorentzian manifold.

UDC: 514.76

Received: 05.12.2011

DOI: 10.4213/mzm9305

 English version: Mathematical Notes, 2013, 93:6, 810–815

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