Natural transformations between \(TTT^*M\) and \(TT^*TM\). (English) Zbl 0806.53024

Doupovec, Miroslav

Czech. Math. J. 43, No. 4, 599-613 (1993).

The author applies the theory of natural geometric operations to the geometry of three times iterated tangent or cotangent bundles. First of all he describes some general properties of the spaces of the form \(HGFM = H(G(F(M))\), where \(M\) is a manifold and \(F\), \(G\), \(H\) are natural bundles. Then he deduces the list of all natural transformations of \(TTT^* M\) into \(TT^* TM\) and interprets all of them geometrically. Further it is shown that all natural operators transforming vector fields on \(M\) into vector fields on \(TT^* M\) can be constructed from the flow operator by applying all natural transformations of \(TTT^* M\) into \(TTT^* M\) over the identity of \(TT^* M\). Finally all natural tensor fields of type \((1,1)\) on \(TT^* M\) are determined.

Reviewer: I.Kolář (Brno)

MSC:

53C05	Connections (general theory)
58A20	Jets in global analysis

Keywords:

iterated bundles; natural transformations; flow operator

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