Summary
Let p: E→B be a fibred manifold. Then, we consider the sheaf\(\mathfrak{B}\)(E)=Ω(B)⊗P(E) of (local) projectable tangent valued forms on E, where Ω(B) is the sheaf of (local) differential forms on B andP(E) is the sheaf of (local) projectable vector fields on E. The Frölicher-Nijenhuis bracket makes\(\mathfrak{B}\)(E) to be a sheaf of graded Lie algebras [18]. In this paper we study all natural R -bilinear operations on\(\mathfrak{B}\)(E) which are of Frölicher-Nijenhuis type. By using the analytical method of [16], we prove that there is a three-parameter family of such operators on\(\mathfrak{B}\)(E). As a consequence, we obtain a result on the unicity of the covariant differential of tangent valued forms and of the curvature associated with a given connection on E. All manifolds and mappings are assumed to be infinitely differentiable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Crampin,Generalized Bianchi identities for horizontal distributions, Math. Proc. Camb. Phil. Soc.,94 (1983), pp. 125–132.
M. Crampin -L. A. Ibort,Graded Lie algebras of derivations and Ehresmann connections, J. Math. Pures Appl.,66 (1987), pp. 113–125.
J. A. Dieudonné -J. B. Carrell,Invariant Theory, Old and New, Academic Press, New York-London, 1971.
T. V. Duc,Sur la géométrie différentielle des fibrés vectoriels, Kōdai Math. Sem. Rep.,26 (1975), pp. 349–408.
C.Ehresmann,Les connexions infinitésimales dans un espace fibré differentiable, Coll. Topologie (Bruxelles, 1950), Liège 1951, pp. 29–55.
A. Frölicher -A. Nijenhuis,Theory of vector valued differential forms, Part 1,Derivations in the graded ring of differential forms, Kon. Ned. Akad. Wet.-Amsterdam, Proc. A 59, Indag. Math.,18 (1956), pp. 338–359.
A. Frölicher -A. Nijenhuis,Theory of vector valued differential forms, Part 2,Almostcomplex structures, Kon. Ned. Akad. Wet.-Amsterdam, Proc. A 61, Indag. Math.,20 (1958), pp. 414–429.
A. Frölicher -A. Nijenhuis,Invariance of vector form operations under mappings, Comm. Math. Hel.,34 (1960), pp. 227–248.
J. Grifone,Structure presque tangent et connexions, I, Ann. Inst. Fourier,22 (1972), pp. 287–334.
J. Janyška,Geometrical properties of prolongation functors, Čas. Pěst. mat.,110 (1985), pp. 77–86.
I. Kolář,Connections in 2-fibered manifolds, Arch. Math. 1, Scripta Fac. Sci. Nat. UJEP Brunensis,17 (1981), pp. 23–30.
I.Kolář,Some natural operators in differential geometry, Proc. Conf. Diff. Geom. and Its Appl., Brno 1986, D. Reidel 1987, pp. 91–110.
I. Kolář,Some natural operations with connections, Journal of National Academy of Mathematics, India5(1987), pp. 129–141.
I.Kolář - P. W.Michor,All natural concomitants of vector values differential forms, Proc. Winter School on Geom. and Phys., Srní 1987, Suppl. Rendiconti Circolo Mat. Palermo, S. II 16 (1987), pp. 101–108.
D. Krupka,Elementary theory of differential invariants, Arch. Math. 4, Scripta Fac. Sci. Nat. UJEP Brunensis,14 (1978), pp. 207–214.
D.Krupka - J.Janyška,Lectures on Differential Invariants, Folia Fac. Sci. Nat. Univ. Purkynianae Brunensis, Brno 1990.
L. Mangiarotti -M. Modugno,Fibered spaces, jet spaces and connections for field theories, in Proc. of Internat. Mett. «Geometry and Physics», Florence 1982, Pitagora Editrice, Bologna 1983, pp. 135–165.
L. Mangiarotti -M. Modugno,Graded Lie algebras and connections on a fibered spaces, J. Math. Pures et Appl.,63 (1984), pp. 111–120.
P. W.Michor,Remarks on the Frölicher-Nijenhuis bracket, Proc. Conf. Diff. Geom. and Its Appl., Brno 1986, D. Reidel, pp. 197–220.
P. W.Michor,Gauge theory for diffeomorphism groups, Proc. Conf. on Diff. Geom. Meth. in Theor. Phys., Como 1987, Kluwer, Dortrecht 1988, pp. 345–371.
M.Modugno,Systems of vector valued forms on a fibred manifold and applications to gauge theories, in Lect. Notes in Math.,1251, Springer-Verlag, 1987.
M. Modugno,Systems of connections and invariant lagrangians, in Differential geometric methods in theoretical physics, Proc. XV Conf. Clausthal 1986, World Publishing, Singapore 1987.
H. K. Nickerson,On differential operators and connections, Trans. Amer. Math. Soc.,99 (1961), pp. 509–539.
A. Nuenhuis,Jacobi-type identities for bilinear differential concomitants of certain tensor fields, Kon. Ned. Akad. Wet.-Amsterdam, Proc. A 58, Indag. Math.,17 (1955), pp. 390–403.
A.Nijenhuis,Vector form methods and deformations of complex structure, Proc. of Symp. in Pure Math. 3, Diff. Geom., Am. Math. Soc., 1961, pp. 87–93.
A.Nijenhuis,Natural bundles and their general properties, Diff. Geom., in honour of K.Yano, Kinokuniya, Tokyo 1972, pp. 317–334.
J, Slovák,On finite order of some operators, Proc. Conf. Diff. Geom. and Its Appl. (communications), Brno 1986, published by the J. E. Purkyně University, Brno 1987, pp. 283–294.
C. L. Terng,Natural vector bundles and natural differential operators, Am. J. Math.,100 (1978), pp. 775–828.
Author information
Authors and Affiliations
Additional information
This paper has been written during the author's visit at the Institute of Applied Mathematics «Giovanni Sansone». Florence, Italy. The author would like to thank ProfessorMarco Modugno for his kind hospitality and for stimulating discussions.
Rights and permissions
About this article
Cite this article
Janyška, J. Natural operations with projectable tangent valued forms on a fibred manifold. Annali di Matematica pura ed applicata 159, 171–187 (1991). https://doi.org/10.1007/BF01766300
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01766300