Previous |  Up |  Next


tangent functor; natural transformations; fibrations; double vector space; double vector fibration; soldering
Our aim is to show a method of finding all natural transformations of a functor $TT^*$ into itself. We use here the terminology introduced in [4,5]. The notion of a soldered double linear morphism of soldered double vector spaces (fibrations) is defined. Differentiable maps $f:C_0\rightarrow C_0$ commuting with $TT^*$-soldered automorphisms of a double vector space $C_0=V^*\times V\times V^*$ are investigated. On the set $Z_s(C_0)$ of such mappings, appropriate partial operations are introduced. The natural transformations $TT^*\rightarrow TT^*$ are bijectively related with the elements of $Z_s((TT^*)_0\bold R^n)$.
[1] I. Kolář: On jet prolongations of smooth categories. Bull. Acad. Polon. Sci., Math., astr. et phys. Vol. XXIV 10 (1976), 883-887. MR 0436190
[2] I. Kolář, Z.Radzisewski: Natural transformations of second tangent and cotangent functors. Czech. Math. Journal 38 (113) (1988), 274-279, Praha. MR 0946296
[3] J. Pradines: Représentation des jets non holonomes par des morphismes vectoriels doubles soudés. C.R.Acad. Sci Paris Sér. A 278 (1974), 1523-1527. MR 0388432 | Zbl 0285.58002
[4] A. Vanžurová: Double vector spaces. Acta Univ. Palac. Olom., Fac. Rer. Nat., Math. XXVI 88 (1987), 9-25. MR 1033327
[5] A. Vanžurová: Double linear connections. AUPO (in press).
[6] A. Vanžurová: Natural transformations of the second tangent functor and soldered morphisms. AUPO, to appear in 1992. MR 1212610
Partner of
EuDML logo