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On naturality of the Helmholtz operator. (English) Zbl 1112.58004

The Helmholtz map in the \(r\)-th order variational calculus on a fibered manifold \(Y\to M\) can be interpreted as a differential operator transforming base preserving morphisms from \(J^{2r} Y\) into \(V^*Y\otimes \bigwedge ^m T^*M\) into base preserving morphisms from \(J^{4r} Y\) into \(V^* J^{2r} Y \otimes V^* Y\otimes \bigwedge ^m T^*M\), \(m= \dim M\).
The author deduces that all natural operators of this type are the constant multiples of the Helmholtz operator.

MSC:

58A20 Jets in global analysis
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