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Curvature tensors whose Jacobi or Szabó operator is nilpotent on null vectors. (English) Zbl 1043.53018
Let \(V\) be a real vector space endowed with a symmetric inner product \((,)\) of signature \((p, q)\). With any algebraic curvature tensor \(R\) on \(V\), for each \(x\in V\) is associated the Jacobi operator \({\mathcal J}_R(x)\), which is defined by: \(({\mathcal J}_R(x) y,w)= R(y,x,x,w)\), for any \(y,w\in V\).
Given a positive integer \(k\leq m-1\), \(m= p+q= \dim V\), let \(\sigma\) be a non-degenerate \(k\)-dimensional subspace of \(V\). The higher-order Jacobi operator is defined as \({\mathcal J}_R(\sigma)= \sum_{1\leq i\leq k} (e_i, e_i){\mathcal J}_R(e_i)\), \(\{e_1,\dots, e_k\}\) being an orthonormal basis of \(\sigma\). The algebraic curvature tensor \(R\) is said to be \(k\)-Osserman if the eigenvalues of \({\mathcal J}_R(\sigma)\) are constant on the Grassmannian of non-degenerate \(k\)-planes in \(V\).
The first part of this paper deals with a detailed investigation of the Jacobi operators. In particular, the authors consider their linear extension to the complexification of \(V\). Using analytic continuation, they prove that, if a tensor \(R\) is \(k\)-Osserman, then any Jacobi operator associated with \(R\) is nilpotent on the set of all complex null vectors. The authors also state that any Lorentzian \(k\)-Osserman manifold has constant sectional curvature. This extends a result already proved for \(k= 1\).
Another relevant space associated with \((V,(,))\) consists of the covariant derivative algebraic curvature tensors on \(V\). If \(T\) is one of such tensors, for any \(x\in V\) the Szabó operator \({\mathcal S}_T(x)\) is defined by: \[ ({\mathcal S}_T(x)y, w)= T(y,x,x,w; x) \] for any \(y,w\in V\).
Useful properties of these operators are known in the Riemannian setting. In this paper the authors study the Szabó operator in the higher-signature setting. In particular, they prove that any covariant derivative algebraic curvature tensor \(T\) on a Lorentzian vector space vanishes, provided that trace \(\{{\mathcal S}_T(\cdot)^2\}\) is constant on the pseudo-spheres of unit timelike and unit spacelike vectors.
On the other hand, if the inner product on \(V\) has signature \((p,q)\), with \(p,q\geq 2\), then there exists a covariant derivative algebraic curvature tensor \(T\) whose Szabó operators don’t vanish and satisfy \({\mathcal S}_T(v)^2= 0\), for any \(v\in V\).

MSC:
53B20 Local Riemannian geometry
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