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Liftings of covariant (0,2)-tensor fields to the bundle of \(k\)-dimensional 1-velocities. (English) Zbl 0905.53010
Slovák, Jan (ed.), Proceedings of the 15th Winter School on geometry and physics, Srní, Czech Republic, January 14–21, 1995. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 43, 111-121 (1996).
The authors first study geometric properties of the lifting of \((0,2)\)-tensor fields on a manifold \(M\) to \((0,2)\)-tensor fields on the bundle \(T^1_kM\) of \(k\)-dimensional one-velocities over \(M\). Then they determine explicitly all first order natural operators of this type.
An important byproduct of this research is an interesting result concerning natural transformations: unlike in the case \(k=1\), there is no natural isomorphism \(T^1_k T^*M\to T^* T^1_kM\) for \(k\geq 2\), where \(T^*\) indicates the cotangent bundle. In conclusion, the authors correct an error in a classification list from a previous paper by M. Doupovec [Arch. Math., Brno 30, 215-225 (1994; Zbl 0816.53007)].
For the entire collection see [Zbl 0884.00042].
Reviewer: I.Kolář (Brno)
53A55 Differential invariants (local theory), geometric objects
53A45 Differential geometric aspects in vector and tensor analysis