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Complete solution of Hadamard’s problem for the scalar wave equation on Petrov type III space-times. (English) Zbl 0951.35131
This very interesting paper is devoted to the solution of Hadamard’s problem on Petrov type III space-times, for the conformally invariant scalar wave equation $\square u+ 1/6 Ru= 0,\tag{1}$ and the non-selfadjoint scalar wave equation $\square u+ A^a\partial_a u+ Cu= 0,$ where $$\square$$ is the Laplace-Beltrami operator corresponding to the metric $$g_{ab}$$ of the background space-time $$V_4$$, $$u$$ is the unknown function, $$R$$ is the Ricci scalar, $$A^a$$ – the components of a given contravariant vector field and $$C$$ – a given scalar function. The background manifold, metric tensor, vector field and scalar function are assumed to be $$C^\infty$$. The main result is that there exists no Petrov type III space-times on which the conformally invariant scalar wave equation (1) satisfies Huygens’ principle. Also there exist no Petrov type III space-times on which the non-selfadjoint scalar wave equation satisfies Huygens’ principle.

##### MSC:
 35Q75 PDEs in connection with relativity and gravitational theory 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 53Z05 Applications of differential geometry to physics
##### Software:
Maple; NP; NPspinor
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