Derivations of differential forms along the tangent bundle projection. (English) Zbl 0748.58002
In J. Phys. A 17, 1999-2009 (1984; Zbl 0542.58011), the third author, F. Cantrijn and M. Crampin introduced a new approach to the characterization of Lagrangian systems, a method which led to a new calculus of certain differential forms on \(T(M)\) [the third author, Differential geometry and its applications, Proc. Conf., Brno/Czech. 1986, Math. Appl., East Eur. Ser. 27, 279-299 (1987; Zbl 0635.58017)]. The present paper introduces a systematic approach to the subject.
Let \(\tau: T(M)\to M\) be the tangent bundle projection and let \(\pi: \Omega(M)\to M\) be the bundle of differential forms. A form along \(\tau\) is a section of the pull-back bundle \(\tau^*\pi\); denote by \(\Lambda(\tau)\) this algebra of sections. First, the authors study the derivations of \(\Lambda(\tau)\): given a connection on \(T(M)\), they obtain classification and characterization of such derivations.
Now, a second-order vector field \(\Gamma\) can be considered as a section \(\gamma\) of \(T^ 2(M)\to T(M)\). The composition of the first prolongation of forms (and fields) along \(\tau\) with the given section \(\gamma\) creates forms (and fields) on \(T(M)\). The authors prove that these objects are the ingredients of the calculus in [the third author, loc. cit.] and use the previous theory of derivations to study them.
Let \(\tau: T(M)\to M\) be the tangent bundle projection and let \(\pi: \Omega(M)\to M\) be the bundle of differential forms. A form along \(\tau\) is a section of the pull-back bundle \(\tau^*\pi\); denote by \(\Lambda(\tau)\) this algebra of sections. First, the authors study the derivations of \(\Lambda(\tau)\): given a connection on \(T(M)\), they obtain classification and characterization of such derivations.
Now, a second-order vector field \(\Gamma\) can be considered as a section \(\gamma\) of \(T^ 2(M)\to T(M)\). The composition of the first prolongation of forms (and fields) along \(\tau\) with the given section \(\gamma\) creates forms (and fields) on \(T(M)\). The authors prove that these objects are the ingredients of the calculus in [the third author, loc. cit.] and use the previous theory of derivations to study them.
Reviewer: D.Tanré (Villeneuve d’Ascq)
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