A common generalization of the Frölicher-Nijenhuis bracket and the Schouten bracket for symmetric multivector fields. (English) Zbl 0844.58002

Dubois-Violette, Michel; Michor, Peter W.

Indag. Math., New Ser. 6, No. 1, 51-66 (1995).

A. Frölicher and A. Nijenhuis [Indag. Math. 18, 338-359 (1956; Zbl 0079.37502)] constructed a generalization of the usual commutator bracket on tangent vector fields to yield a graded Lie algebra of tangent vector-valued forms. This bracket has been applied, for example, in analytical mechanics by C. Godbillon [‘Géometrie differentielle et méchanique analytique’, Paris, Hermann (1969; Zbl 0174.24602)]; and by J. Klein [Ann. Inst. Fourier 12, 1-124 (1962; Zbl 0281.49026)]; to give another definition of connection, by J. Grifone [Ann. Inst. Fourier 22, No. 1, 287-334 (1972; Zbl 0219.53032)] and to study certain Lie subalgebras of the Lie algebra of vector fields which preserve the vertical or/and the horizontal distributions by J. Klein [Colloq. Math. Soc. János Bolyai 46, 665-686 (1988; Zbl 0646.58002)] or a universal version of these results by C. T. J. Dodson and the reviewer [Math. Proc. Camb. Philos. Soc. 103, No. 3, 515-534 (1988; Zbl 0667.53025)]. Another classical extension of the usual bracket of vector fields was defined by J. A. Schouten [Indag. Math. 2, 449-452 (1940; Zbl 0023.17002)] to all symmetric multivector fields. The aim of this paper is to find a common generalization of the Frölicher-Nijenhuis and the Shouten brackets. The authors include a nice example in Riemannian manifolds where this extension would be welcome. The proof of the main theorem is somewhat technical and as the authors observe, their generalization is similar to the one defined by A. M. Vinogradov [Mat. Zametki 47, No. 6, 138-140 (1990; Zbl 0712.58059)].

Reviewer: Lilia Del Riego (San Luis Potosí)

MSC:

58A10	Differential forms in global analysis
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B66	Lie algebras of vector fields and related (super) algebras
17B70	Graded Lie (super)algebras
53C65	Integral geometry

Keywords:

bundle sections; symmetric multivector fields; differential forms

Citations:

Zbl 0079.37502; Zbl 0174.24602; Zbl 0281.49026; Zbl 0219.53032; Zbl 0646.58002; Zbl 0667.53025; Zbl 0023.17002; Zbl 0712.58059

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