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The first order PDE system for type III Osserman manifolds. (English) Zbl 0949.53016

In the case of a pseudo-Riemannian manifold \(M\) we say that \(M\) is a spacelike (resp. timelike) Osserman manifold at point \(p\in M\) if the Jordan form of \(K_X\) is independent of \(X\in T_pM\) at \(p\in M\), where \(X\) is a unit spacelike (timelike) tangent vector. Here, \(K_X\) denotes the restriction of the Jacobi operator \(R_X: Y\mapsto R(Y,X)X\) to \(X^\perp\), where \(T_pM=X\oplus X^\perp\) and \(X\) is a non-null tangent vector at \(p\in M\).
The authors investigate \(4\)-dimensional pseudo-Riemannian manifolds of signature \((2,2)\). In this case, the Osserman condition is equivalent to the independence of the minimal polynomial of \(K_X\) from \(X\). The main purpose of this paper is to study the special case when the minimal polynomial of \(K_X\) has a triple zero. The examples in this special case reveal that the geometry of such manifolds is very rich and interesting. In the most general case, the Osserman type condition leads to a system of PDEs of second order. As a consequence of this system, using the second Bianchi identity, the authors reduce the problem to a more natural and simpler PDE system of first order (in terms of the connection forms of a specific null frame). Sometimes, this system is sufficient to establish the characterization or to prove the nonexistence of the corresponding manifolds.

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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