On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska


We describe all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(D\colon Q^{\tau}_{proj-proj} \rightsquigarrow QT^*\) transforming projectable-projectable classical torsion-free linear connections \(\nabla\) on fibred-fibred manifolds \(Y\) into classical linear connections \(D(\nabla)\) on cotangent bundles \(T^*Y\) of \(Y\). We show that this problem can be reduced to finding \(\mathcal{F}^2 \mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(D\colon Q^{\tau}_{proj-proj}\rightsquigarrow
(T^*,\otimes^pT^*\otimes\otimes^q T)\) for \(p=2\), \(q=1\) and \(p=3\), \(q=0\).


Fibred-fibred manifold; projectable-projectable linear connection; natural operator.

Full Text:



Doupovec, M., Mikulski, W. M., On prolongation of higher order onnections, Ann. Polon. Math. 102, no. 3 (2011), 279–292.

Kolar, I., Connections on fibered squares, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 59 (2005), 67–76.

Kolar, I., Michor, P. W., Slovak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin–Heidelberg, 1993.

Kurek, J., Mikulski, W. M., On prolongations of projectable connections, Ann. Polon. Math. 101, no. 3 (2011), 237–250.

Kurek, J., Mikulski, W. M., The natural liftings of connections to tensor powers of the cotangent bundle, AGMP-8 Proceedings (Brno 2012), Miskolc Mathematical Notes, to appear.

Kures, M., Natural lifts of classical linear connections to the cotangent bundle, Suppl. Rend. Mat. Palermo II 43 (1996), 181–187.

Mikulski, W. M., The jet prolongations of fibered-fibered manifolds and the flow operator, Publ. Math. Debrecen 59 (3–4) (2001), 441–458.

Yano, K., Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker, Inc., New York, 1973.

DOI: http://dx.doi.org/10.17951/a.2013.67.1.1-10
Data publikacji: 2015-07-15 00:00:00
Data złożenia artykułu: 2016-01-11 18:04:49


Total abstract view - 267
Downloads (from 2020-06-17) - PDF - 147



  • There are currently no refbacks.

Copyright (c) 2016 Anna Bednarska