Abstract
We give a classification of all linear natural operators transforming p-vectors (i.e., skew-symmetric tensor fields of type (p, 0)) on n-dimensional manifolds M to tensor fields of type (q, 0) on T A M, where T A is a Weil bundle, under the condition that p ≥ 1, n ≥ p and n ≥ q. The main result of the paper states that, roughly speaking, each linear natural operator lifting p-vectors to tensor fields of type (q, 0) on T A is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting p-vectors to tensor fields of type (p, 0) on T A and canonical tensor fields of type (q − p, 0) on T A.
Similar content being viewed by others
References
J. Dębecki: Canonical tensor fields of type (p, 0) on Weil bundles. Ann. Pol. Math. 88 (2006), 271–278.
J. Dębecki: Linear liftings of skew-symmetric tensor fields to Weil bundles. Czech. Math. J. 55 (2005), 809–816.
D. J. Eck: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133–140.
J. Grabowski, P. Urbański: Tangent lifts of Poisson and related structures. J. Phys. A, Math. Gen. 28 (1995), 6743–6777.
G. Kainz, P. W. Michor: Natural transformations in differential geometry. Czech. Math. J. 37 (1987), 584–607.
I. Kolář: Weil Bundles as Generalized Jet Spaces. Handbook of Global Analysis (D. Krupka et al., eds.). Elsevier, Amsterdam, 2008, pp. 625–664.
I. Kolář: On the natural operators on vector fields. Ann. Global Anal. Geom. 6 (1988), 109–117.
I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry. (corrected electronic version), Springer, Berlin, 1993.
O. O. Luciano: Categories of multiplicative functors and Weil’s infinitely near points. Nagoya Math. J. 109 (1988), 69–89.
W. M. Mikulski: The linear natural operators lifting 2-vector fields to some Weil bundles. Note Mat. 19 (1999), 213–217.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dębecki, J. Linear natural operators lifting p-vectors to tensors of type (q, 0) on Weil bundles. Czech Math J 66, 511–525 (2016). https://doi.org/10.1007/s10587-016-0272-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-016-0272-z