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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
##### Keywords:
curvature tensors; Jacobi operators; Szabó operators
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