Natural transformations transforming vector fields into affinors on the extended \(r\)-th order tangent bundles. (English) Zbl 0804.58005

Mikulski, Włodzimierz M.

Arch. Math., Brno 29, No. 1-2, 59-70 (1993).

For any \(n\)-dimensional manifold \(M\) supposed to be of class \(C^ \infty\), the extended \(r\)-th order tangent bundle \(E^ r M\) over \(M\) is defined by \(E^ r M = (J^ r (M,R))^*\). The author determines all natural transformations transforming vector fields on \(n\)-dimensional manifolds into affinors (i.e. tensor fields of type (1.1)) on \(E^ r\). For this purpose, the author defines geometrically \(2(r + 2)\) natural transformations transforming vector fields on \(n\)-dimensional manifolds into affinors on \(E^ r\), and prove that all natural transformations transforming vector fields on \(n\)-dimensional manifolds into affinors on \(E^ r\) are their linear combinations, the coefficients of which are arbitrary smooth functions on \(R\), provided \(n \geq 3\).

Reviewer: N.Papaghiuc (Iaşi)

MSC:

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

Keywords:

natural transformations; vector fields; affinors