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A global characterization of jet bundles of \(p^ 1\)-velocities and covelocities. (English) Zbl 0846.53019
The authors first give the notion of \(p\)-almost tangent structures on a smooth \((p + 1)n\)-dimensional manifold \(M\) and give the global characterization of the tangent bundle of \(p^1\)-velocities in terms of the integrability of \(p\)-almost tangent structures. Next, the authors give similar notions of \(p\)-almost cotangent bundles and the global characterization of cotangent bundles of \(p^1\)-covelocities. For the proofs the authors use the characterization of vector bundles by T. Nagano [J. Differ. Geom. 2, 253-264 (1968; Zbl 0172.46903)].
MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A20 Jets in global analysis
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[1] Awane ( A. ) .- Sur une généralisation des structures symplectiques , Thèse de 3ème cycle , Publications de l’Institut de Recherche Mathématique Avancée , Strasbourg , 1984 .
[2] Awane ( A. ) .- k-Symplectic structures , J. Math. Phys. 33 , n^\circ 12 ( 1992 ), pp. 4046 - 4052 . MR 1191763 | Zbl 0781.53024 · Zbl 0781.53024
[3] Awane ( A. ) . - Some affine properties of k-symplectic structures , Preprint. MR 1614431 · Zbl 0908.53018
[4] Crampin ( M. ) and Thompson ( G. ) .- Affine bundles and integrable almost tangent structures , Math. Proc. Camb. Phil. Soc. 98 ( 1985 ), pp. 61 - 71 . MR 789719 | Zbl 0572.53031 · Zbl 0572.53031
[5] De Filippo ( S. ), Landi ( G. ), Marmo ( G. ) and Vilasi ( G. ) .- Tensor fields defining a tangent bundle structure , Ann. Inst. Henri-Poincaré, Physique théorique 50 , n^\circ 2 ( 1989 ), pp. 205 - 218 . Numdam | MR 1002820 | Zbl 0752.58010 · Zbl 0752.58010
[6] De León ( M. ), Méndez ( I. ) and Salgado ( M. ) .- Regular p-almost cotangent structure , J. Korean Math. Soc. 25 , n^\circ 2 ( 1988 ), pp. 273 - 287 . MR 968124 | Zbl 0657.53021 · Zbl 0657.53021
[7] De León ( M. ), Méndez ( I. ) and Salgado ( M. ) .- Integrable p-almost tangent manifolds and tangent bundles of p1-velocities , Acta Mathematica Hungarica 58 , n^\circ 1 -2 ( 1991 ), pp. 45 - 54 . MR 1152825 | Zbl 0757.53015 · Zbl 0757.53015
[8] De León ( M. ), Méndez ( I. ) and Salgado ( M. ) .- p-Almost tangent structures , Rend. Circ. Mat. Palermo , Serie II , XXXVII ( 1988 ), pp. 282 - 294 . MR 994618 | Zbl 0672.53040 · Zbl 0672.53040
[9] De León ( M. ), Méndez ( I. ) and Salgado ( M. ) .- p-Almost cotangent structures , Bollettino U. M. I. 7 - A , n^\circ 7 ( 1993 ), pp. 97 - 107 . MR 1215103 | Zbl 0777.53033 · Zbl 0777.53033
[10] De León ( M. ), Merino ( E. ), Oubiña ( J.A. ) and Salgado ( M. ) .- A characterization of tangent and stable tangent bundles , to appear in Ann. Inst. Henri-Poincaré, Physique Théorique , 61 , n^\circ 1 ( 1994 ). Numdam | MR 1303183 | Zbl 0815.53039 · Zbl 0815.53039
[11] De León ( M. ) and Rodrigues ( P.R. ) .- Generalized Classical Mechanics and Field Theory , North-Holland Math. , Serie 112 , Amsterdam ( 1985 ). MR 808964 | Zbl 0581.58015 · Zbl 0581.58015
[12] De León ( M. ) and Rodrigues ( P.R. ) .- A contribution to the global formulation of the higher Poincaré-Cartan form , Letters in Mathematical Physics 14 , n^\circ 4 ( 1987 ), pp. 353 - 362 . MR 922830 | Zbl 0637.53056 · Zbl 0637.53056
[13] De León ( M. ) and Rodrigues ( P.R. ) .- Hamiltonian structures and Lagrangian field theories on jet bundles , Bol. Acad. Galega de Ciencias VII ( 1988 ), pp. 69 - 81 .
[14] De León ( M. ) and Rodrigues ( P.R. ) .- nk-Almost Tangent Structures and the Hamiltonization of Higher Order Field Theories , J. Math. Phys. 30 , n^\circ 6 ( 1989 ), pp. 1351 - 1353 . MR 995781 | Zbl 0689.53018 · Zbl 0689.53018
[15] De León ( M. ) and Rodrigues ( P.R. ) .- Methods of Differential Geometry in Analytical Mechanics , North-Holland Math. , Serie 152 , Amsterdam ( 1989 ). MR 1021489 | Zbl 0687.53001 · Zbl 0687.53001
[16] Méndez ( I. ) .- p-Estructuras casi tangentes y p-estructuras casi cotangentes , Publicaciones del Departamento de Geometría y Topología 78 , Universidad de Santiago de Compostela ( 1990 ). Zbl 0839.53022 · Zbl 0839.53022
[17] Morimoto ( A. ) . - Prolongations of Geometric Structures , Lecture Note , Math. Inst. Nagoya Univ. ( 1969 ). MR 276893 | Zbl 0223.53027 · Zbl 0223.53027
[18] Nagano ( T. ) .- 1-Forms with the exterior derivative of maximal rank , J. Differential Geometry 2 ( 1968 ), pp. 253 - 264 . MR 239526 | Zbl 0172.46903 · Zbl 0172.46903
[19] Thompson ( G. ) . - Integrable almost cotangent structures and Legendrian bundles , Math. Proc. Camb. Phil. Soc. 101 ( 1987 ), pp. 61 - 78 . MR 877701 | Zbl 0629.53034 · Zbl 0629.53034
[20] Thompson ( G. ) and Schwardmann ( U. ) .- Almost tangent and cotangent structures in the large , Transactions of the Amer. Math. Soc. 327 , n^\circ 1 ( 1991 ), pp. 313 - 328 . MR 1012509 | Zbl 0737.53035 · Zbl 0737.53035
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