Natural transformations of affinors into functions and affinors. (English) Zbl 0816.58002

Bureš, J. (ed.) et al., The proceedings of the 11th winter school on geometry and physics held in Srní, Czechoslovakia, January 5-12, 1991. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 30, 101-112 (1993).
The author’s main results can be formulated as follows:
(1) All natural transformations of affinors (i.e. tensor fields of type (1,1)) on \(n\)-dimensional manifolds into functions are of the form \(F(a_ 1(t),\dots,a_ n(t))\), where \(a_ 1(t),\dots,a_ n(t)\) denote the coefficients of the characteristic polynomial of the affinor \(t\) and \(F\) is a smooth function on \(\mathbb{R}^ n\).
(2) All natural transformations of affinors on an \(n\)-manifold \(M\) into \({\mathfrak H}^ 1_ 1(M)\) are of the form \[ t \in {\mathfrak H}^ 1_ 1 (M) \mapsto \sum^ n_{k = 1} F_ k(a_ 1(t),\dots,a_ n(t)) t^{n- k}, \] where \(F_ 1,\dots,F_ n\) are smooth functions on \(\mathbb{R}^ n\).
For the entire collection see [Zbl 0777.00026].


58A20 Jets in global analysis
53A55 Differential invariants (local theory), geometric objects