Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

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December 2012, 4(4): 397-419. doi: 10.3934/jgm.2012.4.397

Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

Arturo Echeverría-Enríquez ^1, , Alberto Ibort ^2, , Miguel C. Muñoz-Lecanda ^3, and Narciso Román-Roy ^4,

1.	Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona, Spain
2.	Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA, and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
3.	Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona, Spain
4.	Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona

Received May 2012 Revised October 2012 Published January 2013

Abstract
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It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree $> 1$, which is locally homogeneous of degree $k$ with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

Keywords: graded Lie algebras., invariant forms, Multisymplectic manifolds, Hamiltonian (multi) vector fields, multisymplectic diffeomorphisms.

Mathematics Subject Classification: Primary: 53C15, 53D35; Secondary: 57R50, 57S25, 58A1.

Citation: Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397

References:

[1]	C. J. Atkin and J. Grabowsk, Homomorphisms of the Lie algebras associated with a symplectic manifold, Comp. Math., 76 (1990), 315-349.
[2]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.
[3]	A. Banyaga, On isomorphic classical diffeomorphism groups. I, Proc. Am. Math. Soc., 98 (1986), 113-118. doi: 10.2307/2045779.
[4]	A. Banyaga, On isomorphic classical diffeomorphism groups. II, J. Diff. Geom., 28 (1988), 23-35.
[5]	A. Banyaga, The structure of classical diffeomorphism groups, in "Mathematics and Its Applications," 400, Kluwer Acad. Pub. Group., Dordrecht, (1997), 113-118.
[6]	A. Banyaga and A. McInerney, On isomorphic classical diffeomorphism groups. III, Ann. Global Anal. Geom., 13 (1995), 117-127. doi: 10.1007/BF01120327.
[7]	W. M. Boothby, Transitivity of the automorphisms of certain geometric structures, Amer. Math. Soc., 137 (1969), 93-100.
[8]	R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. (2), 126 (1987), 525-576. doi: 10.2307/1971360.
[9]	F. Cantrijn, A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236.
[10]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser., 66 (1999), 303-330.
[11]	J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.
[12]	J. F. Cariñena, J. Gomis, L. A. Ibort and N. Román-Roy, Canonical transformation theory for presymplectic systems, J. Math. Phys., 26 (1985), 1961-1969. doi: 10.1063/1.526864.
[13]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603. doi: 10.1063/1.532525.
[14]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484. doi: 10.1088/0305-4470/32/48/309.
[15]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[16]	A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901 (30 pp). doi: 10.1063/1.2801875.
[17]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Sci. Pub. Co., Singapore, 1997.
[18]	J. Gomis, J. Llosa and N. Román-Roy, Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems, J. Math. Phys., 25 (1984), 1348-1355. doi: 10.1063/1.526303.
[19]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (ed. M. Francaviglia), Elsevier Science Pub., (1991), 203-235.
[20]	J. Grabowski, "Isomorphisms of Poisson and Jacobi Brackets," Banach Center Publ., 51, Polish Acad. Sci., Warsaw, (2000), 79-85.
[21]	F. Helein and J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43 (2002), 2306-2347. doi: 10.1063/1.1467710.
[22]	L. Hwa Chung, The universal integral invariants of Hamiltonian systems and applications to the theory of canonical transformations, Proc. Roy. Soc., LXIIA (1947), 237-246.
[23]	L. A. Ibort, Multisymplectic geometry: Generic and exceptional, in "IX Fall Workshop on Geometry and Physics, Vilanova i la Geltrú, Spain (2000)" (eds. X. Gràcia, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy), UPC Eds., (2001), 79-88.
[24]	I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90. doi: 10.1016/S0034-4877(98)80182-1.
[25]	J. Kijowski and W. M. Tulckzyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics 107, Springer-Verlag, New York, 1979.
[26]	M. de León D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, in "Applied Differential Geometry and Mechanics" (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47.
[27]	M. de León, D. Martín de Diego and A. Santamaría-Merino, Symmetries in classical field theories, Int. J. Geom. Meth. Mod. Phys., 1 (2004), 651-710. doi: 10.1142/S0219887804000290.
[28]	J. Llosa and N. Román-Roy, Invariant forms and Hamiltonian systems: A geometrical setting, Int. J. Theor. Phys., 27 (1988), 1533-1543. doi: 10.1007/BF00669290.
[29]	C. M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys., 23 (1997), 350-359. doi: 10.1016/S0393-0440(97)80009-5.
[30]	J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575. doi: 10.1017/S0305004198002953.
[31]	J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier, Grenoble, 20 (1970), 95-178.
[32]	H. Omori, "Infinite Dimensional Lie Transformation Groups," Lect. Notes in Maths., 427, Springer-Verlag, Berlin and New York , 1974.
[33]	C. Paufler and H. Romer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69. doi: 10.1016/S0393-0440(02)00031-1.
[34]	L. E. Pursell and M. E. Shanks, The Lie algebra of smooth manifolds, Proc. Am. Math. Soc., 5 (1954), 468-472.
[35]	N. Román-Roy, A. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137.
[36]	G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory," World Scientific Pub., Singapore, 1995. doi: 10.1142/9789812831484.
[37]	D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989. doi: 10.1017/CBO9780511526411.
[38]	M. Shafiee, On Hamiltonian group of multisymplectic manifolds, Int. J. Geom. Meth. Mod. Phys., 8 (2011), 929-935. doi: 10.1142/S0219887811005506.
[39]	F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol Soc. Brasil. Mat., 10 (1979), 17-25. doi: 10.1007/BF02588337.
[40]	W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Hamiltoniènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 15-18.
[41]	W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Lagrangiènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 675-678.
[42]	M. Wechsler, Homeomorphism groups of certain topological spaces, Ann. Math., 62 (1954), 360-373.

show all references

References:

[1]	C. J. Atkin and J. Grabowsk, Homomorphisms of the Lie algebras associated with a symplectic manifold, Comp. Math., 76 (1990), 315-349.
[2]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.
[3]	A. Banyaga, On isomorphic classical diffeomorphism groups. I, Proc. Am. Math. Soc., 98 (1986), 113-118. doi: 10.2307/2045779.
[4]	A. Banyaga, On isomorphic classical diffeomorphism groups. II, J. Diff. Geom., 28 (1988), 23-35.
[5]	A. Banyaga, The structure of classical diffeomorphism groups, in "Mathematics and Its Applications," 400, Kluwer Acad. Pub. Group., Dordrecht, (1997), 113-118.
[6]	A. Banyaga and A. McInerney, On isomorphic classical diffeomorphism groups. III, Ann. Global Anal. Geom., 13 (1995), 117-127. doi: 10.1007/BF01120327.
[7]	W. M. Boothby, Transitivity of the automorphisms of certain geometric structures, Amer. Math. Soc., 137 (1969), 93-100.
[8]	R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. (2), 126 (1987), 525-576. doi: 10.2307/1971360.
[9]	F. Cantrijn, A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236.
[10]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser., 66 (1999), 303-330.
[11]	J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.
[12]	J. F. Cariñena, J. Gomis, L. A. Ibort and N. Román-Roy, Canonical transformation theory for presymplectic systems, J. Math. Phys., 26 (1985), 1961-1969. doi: 10.1063/1.526864.
[13]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603. doi: 10.1063/1.532525.
[14]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484. doi: 10.1088/0305-4470/32/48/309.
[15]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[16]	A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901 (30 pp). doi: 10.1063/1.2801875.
[17]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Sci. Pub. Co., Singapore, 1997.
[18]	J. Gomis, J. Llosa and N. Román-Roy, Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems, J. Math. Phys., 25 (1984), 1348-1355. doi: 10.1063/1.526303.
[19]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (ed. M. Francaviglia), Elsevier Science Pub., (1991), 203-235.
[20]	J. Grabowski, "Isomorphisms of Poisson and Jacobi Brackets," Banach Center Publ., 51, Polish Acad. Sci., Warsaw, (2000), 79-85.
[21]	F. Helein and J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43 (2002), 2306-2347. doi: 10.1063/1.1467710.
[22]	L. Hwa Chung, The universal integral invariants of Hamiltonian systems and applications to the theory of canonical transformations, Proc. Roy. Soc., LXIIA (1947), 237-246.
[23]	L. A. Ibort, Multisymplectic geometry: Generic and exceptional, in "IX Fall Workshop on Geometry and Physics, Vilanova i la Geltrú, Spain (2000)" (eds. X. Gràcia, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy), UPC Eds., (2001), 79-88.
[24]	I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90. doi: 10.1016/S0034-4877(98)80182-1.
[25]	J. Kijowski and W. M. Tulckzyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics 107, Springer-Verlag, New York, 1979.
[26]	M. de León D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, in "Applied Differential Geometry and Mechanics" (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47.
[27]	M. de León, D. Martín de Diego and A. Santamaría-Merino, Symmetries in classical field theories, Int. J. Geom. Meth. Mod. Phys., 1 (2004), 651-710. doi: 10.1142/S0219887804000290.
[28]	J. Llosa and N. Román-Roy, Invariant forms and Hamiltonian systems: A geometrical setting, Int. J. Theor. Phys., 27 (1988), 1533-1543. doi: 10.1007/BF00669290.
[29]	C. M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys., 23 (1997), 350-359. doi: 10.1016/S0393-0440(97)80009-5.
[30]	J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575. doi: 10.1017/S0305004198002953.
[31]	J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier, Grenoble, 20 (1970), 95-178.
[32]	H. Omori, "Infinite Dimensional Lie Transformation Groups," Lect. Notes in Maths., 427, Springer-Verlag, Berlin and New York , 1974.
[33]	C. Paufler and H. Romer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69. doi: 10.1016/S0393-0440(02)00031-1.
[34]	L. E. Pursell and M. E. Shanks, The Lie algebra of smooth manifolds, Proc. Am. Math. Soc., 5 (1954), 468-472.
[35]	N. Román-Roy, A. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137.
[36]	G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory," World Scientific Pub., Singapore, 1995. doi: 10.1142/9789812831484.
[37]	D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989. doi: 10.1017/CBO9780511526411.
[38]	M. Shafiee, On Hamiltonian group of multisymplectic manifolds, Int. J. Geom. Meth. Mod. Phys., 8 (2011), 929-935. doi: 10.1142/S0219887811005506.
[39]	F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol Soc. Brasil. Mat., 10 (1979), 17-25. doi: 10.1007/BF02588337.
[40]	W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Hamiltoniènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 15-18.
[41]	W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Lagrangiènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 675-678.
[42]	M. Wechsler, Homeomorphism groups of certain topological spaces, Ann. Math., 62 (1954), 360-373.

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