One component of the curvature tensor of a Lorentzian manifold
The basic local invariant of a (pseudo)-Riemannian manifold \((M^n,g)\) is the Riemannian curvature tensor \(R\). The values of \(R\) at a point are contained in the holonomy algebra \(\mathfrak{g}\) at that point. In Riemannian signature the algebraic spaces \(\mathcal{R}(\mathfrak{g})\) of curvature tensors of type \(\mathfrak{g}\) are known for any \(\mathfrak{g}\). The present article aims to compute the \(\mathcal{R}(\mathfrak{g})\) for the holonomy algebras \(\mathfrak{g}\) of a Lorentzian manifold. In the Lorentzian case \(\mathcal{R}(\mathfrak{g})\) is derived from \(\mathcal{R}(\mathfrak{h})\) and \(\mathcal{P}(\mathfrak{h})\), where \(\mathfrak{h}\subset\mathfrak{so}(n)\) is the projection of \(\mathfrak{g}\subset\mathfrak{so}(1,n+1)\) to \(\mathfrak{so}(n)\). This again is known to be a holonomy algebra of a Riemannian manifold, i.e., a Berger algebra. Hence, \(\mathcal{R}(\mathfrak{h})\) is known and it remains to compute the component \[ \mathcal{P}(\mathfrak{h})\;:=\{P\in(\mathbb{R}^n)^*\otimes \mathfrak{h}\;|\;(P(u)v,w)+(P(v)w,u)+(P(w)u,v)=0\;\forall u,v,w\in\mathbb{R}^n\}. \] Going through the list of irreducible Berger subalgebras \(\mathfrak{h}\) of \(\mathfrak{so}(n,\mathbb{C})\), the author calculates the spaces \(\mathcal{P}(\mathfrak{h})\) explicitly as \(\mathfrak{h}\)-modules. In particular, he obtaines the \(\mathcal{P}(\mathfrak{h})\) for the symmetric Berger algebras and the exceptional algebras \(\mathfrak{spin}(7)\) and \(\mathfrak{g}_2\). The case \(\mathfrak{h}\subset\mathfrak{u}(n/2)\) was known before already.
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