Natural 2-forms on the tangent bundle of a Riemannian manifold. (English) Zbl 0818.53020

Bureš, J. (ed.) et al., The proceedings of the winter school geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 165-174 (1994).
It is known that there is a canonical 2-form on the cotangent bundle \(T^* M\to M\), the symplectic structure, and the existence of a canonical 2-form on the tangent bundle \(TM\to M\) of a Riemannian manifold easily follows. This provides an example of a natural 2-form of order 1 on \(TM\), and the author determines all such 2-forms by direct calculation. Together with O. Kowalski and M. Sekizawa [Bull. Tokyo Gakugei Univ., Sect. IV., Math. Nat. Sci. 40, 1-29 (1988; Zbl 0656.53021)], this gives the complete classification of natural \((0,2)\)- tensor fields on \(TM\). The result is applied to show some examples of natural transformations \(TTM\to T^* TM\) for Riemannian manifolds.
For the entire collection see [Zbl 0794.00022].
Reviewer: J.Chrastina (Brno)


53B20 Local Riemannian geometry


Zbl 0656.53021