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Natural transformations of second tangent and cotangent functors. (English) Zbl 0669.53023
The paper classifies all natural transformations between the two functors \(TT^*\) and \(T^*T\), by showing that they are parametrized by three arbitrary smooth functions of one variable. Among them, a distinguished transformation s: \(TT^* \to T^*T\) is characterized geometrically. This situation is analogous to that concerning the functor TT. In fact, there is a four parameter family of natural transformations of the functor TT into itself and the distinguished involution s: TT\(\to TT\) can be characterized geometrically.
Moreover, by considering the symplectic structure of the cotangent bundle (which can be viewed as a natural transformation \(TT^*\to T^*T^*)\), the authors deduce that all the natural transformations between any two of the functors \(TT^*\), \(T^*T\) and \(T^*T^*\) depend on three arbitrary smooth functions of one variable. Eventually, the authors explain (in terms of the Weil algebra) the deep reason by which the functor TT is not naturally related to the other three functors \(TT^*\), \(T^*T\) and \(T^*T^*\). This fact is in accordance with the lack of a natural symplectic structure on the tangent bundle.
Reviewer: M.Modugno

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A99 General theory of differentiable manifolds
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References:
[1] C. Godbillon: Géométrie différentielle et méchanique analytique. Paris 1969. · Zbl 0174.24602
[2] /. Janyska: Geometrical properties of prolongation functors. Čas. pro pěst. mat. 110 (1985), 77-86. · Zbl 0582.58002
[3] G. Kainz P. Michor: Natural transformations in differential geometry. to appear in Czech. Math. J. · Zbl 0654.58001
[4] I. Kolář: On the second tangent bundle and generalized Lie derivatives. Tensor, N.S., 38 (1982), 98-102.
[5] I. Kolář: Natural transformations of the second tangent functor into itself. Arch. Math. (Brno), XX(1984), 169-172. · Zbl 0578.58004
[6] I. Kolář Z. Radziszewski: Some \(GL(n,R)\)-equivariant smooth maps. Seminar report, Brno 1986.
[7] M. Modugno G. Stefani: Some results on second tangent and cotangent spaces. Quaderni dell’Istituto di Matematica dell’ Università di Lecce, Q. 16, 1978.
[8] A. Nijenhuis: Natural bundles and their general properties. Differential Geometry, in honour of K. Yano, Kinokuniya, Tokyo, 1972, 317-334. · Zbl 0246.53018
[9] R. S. Palais C. L. Terng: Natural bundles have finite order. Topology 16 (1977), 271 - 277. · Zbl 0359.58004
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