Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T17:12:06.364Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometry of multisymplectic manifolds

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

F. Cantrijn ,
A. Ibort  and
M. De León
F. Cantrijn
Affiliation:
Theoretical Mechanics Division, University of Gent, Krijgslaan 281, B-9000 Gent, Belgium
A. Ibort
Affiliation:
Departmento de Matematicas, Universidad Carlos III de Madrid, E-28911 Leganes, Madrid, Spain
M. De León
Affiliation:
Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, E-28006 Madrid, Spain
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A multisymplectic structure on a manifold is defined by a closed differential form with zero characteristic distribution. Starting from the linear case, some of the basic properties of multisymplectic structures are described. Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a bundle of exterior k-forms on a manifold. For a class of multisymplectic manifolds admitting a ‘Lagrangian’ fibration, a general structure theorem is given which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.

MSC classification

Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 3 , June 1999 , pp. 303 - 330
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Abraham, R. and Marsden, J. E., Foundations of mechanics (Benjamin Cummings, Reading, MA, 1978).Google Scholar
[2]Awane, A., ‘k-symplectic structures’, J. Math. Phys. 33 (1992), 4046–4052.CrossRefGoogle Scholar
[3]Awane, A., ‘G-espaces K-symplectiques homogènes’, J. Geom. Phys. 13 (1994), 139–157.CrossRefGoogle Scholar
[4]Binz, E., Śniatycki, J. and Fisher, H., Geometry of classical fields, Math. Studies 154 (North-Holland, Amsterdam, 1989).Google Scholar
[5]Bonan, E., ‘Sur l'algèbre extérieure d'une variété hermitienne quaternionique’, C. R. Acad. Sci. Paris (Série l) 295 (1982), 115–118.Google Scholar
[6]Cariñena, J. F., Crampin, M. and Ibort, L. A., ‘On the multisymplectic formalism for first order field theories’, Differential Geom. Appl. 1 (1991), 345–374.CrossRefGoogle Scholar
[7]Fulp, R. O., Lawson, J. K. and Norris, L. K., ‘Generalized symplectic geometry as a covering theory for Hamiltonian theories of classical particles and fields’, J. Geom. Phys. 20 (1996), 195–206.CrossRefGoogle Scholar
[8]Garcia, P. L. and Perez-Rendon, A., ‘Symplectic approach to the theory of quantized fields II’, Arch. Rational Mech. Anal. 43 (1971), 101–124.CrossRefGoogle Scholar
[9]Goldschmidt, H. and Sternberg, S., ‘The Hamilton-Cartan formalism in the calculus of variations’, Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267.CrossRefGoogle Scholar
[10]Gotay, M. J., ‘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. Francaviglia, M.) (Elsevier, New York, 1991) pp. 203–235.CrossRefGoogle Scholar
[11]Gotay, M. J., ‘A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition’, Differential Geom. Appl. 1 (1991), 375–390.CrossRefGoogle Scholar
[12]Gotay, M., Isenberg, J., Marsden, J. E. and Montgomery, R., ‘Momentum mappings and the Hamiltonian structure of classical field theories with constraints’, Technical report.Google Scholar
[13]Günther, C., ‘The polysymplectic Hamiltonian formalism in field theories and the calculus of variations’, J. Differential Geom. 25 (1987), 23–53.CrossRefGoogle Scholar
[14]Hitchin, N. J., ‘The self-duality equations on a Riemann surface’, Proc. London Math. Soc. 55 (1987), 59–126.CrossRefGoogle Scholar
[15]Ibort, A., ‘The geometry of moduli spaces of Yang-Mills theories in dimension 3 and 4’, An. de Fisica, Monografias RSEF 3 (1995), 151–165.Google Scholar
[16]Kanatchikov, I., ‘Novel algebraic structures from the polysymplectic form in field theory’, in: GROUP 21, Physical applications and mathematical aspects of geometry, groups and algebras, vol. 2 (eds. Doebner, H.-D., Scherer, W. and Schulte, C.) (World Scientific, London, 1997) pp. 894–899.Google Scholar
[17]Kijowski, J. and Szczyrba, W., ‘Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory’, in: Géométrie symplectique et physique mathématique, Coll. Int. C.N.R.S. No 237 (C.N.R.S., 1975) pp. 347–378.Google Scholar
[18]Kijowski, J., ‘A finite-dimensional canonical formalism in the classical field theory’, Commun. Math. Phys. 30 (1973), 99–128.CrossRefGoogle Scholar
[19]Kijowski, J. and Tulczyjew, W. M., A symplectic framework for field theories, Lecture Notes in Phys. 107 (Springer, Berlin, 1979).CrossRefGoogle Scholar
[20]de León, M., Mendez, I. and Salgado, M., ‘p-almost cotangent structures’, Boll. Un. Mat. Ital. A-X7 (1993), 97–107.Google Scholar
[21]Libermann, P. and Marle, C.-M., Symplectic geometry and analytical mechanics, Mathematics and its Applications (D. Reidel, Boston, 1987).CrossRefGoogle Scholar
[22]Martin, G., ‘A Darboux theorem for multi-symplectic manifolds’, Lett. Math. Phys. 16 (1988), 133–138.CrossRefGoogle Scholar
[23]Martin, G., ‘Dynamical structures for k-vector fields’, Internat. J. Theoret. Phys. 27 (1988), 571–585.CrossRefGoogle Scholar
[24]Martinet, J., ‘Sur les singularités des formes différentielles’, Ann. Inst. Fourier (Grenoble) 20 (1970), 95–178.CrossRefGoogle Scholar
[25]Nagano, T., ‘1-forms with the exterior derivative of maximal rank’, J. Differential Geom. 2 (1968), 253–264.CrossRefGoogle Scholar
[26]Norris, L. K., ‘Generalized symplectic geometry on the frame bundle of a manifold’, Proc. Sympos. Pure Math. 54 (1993), 435–465.CrossRefGoogle Scholar
[27]Revoy, P., ‘Trivecteurs de rang 6’, Bull. Soc. Math. France. Mem. 59 (1979), 141–155.Google Scholar
[28]Sardanashvily, G., Generalized Hamiltonian formalism for field theory. Constraint Systems (World Scientific, London, 1995).CrossRefGoogle Scholar
[29]Sardanashvily, G. and Zakharov, O., ‘On application of the Hamiltonian formalism in fibred manifolds to field theory’, Differential Geom. Appl. 3 (1993), 245–263.CrossRefGoogle Scholar
[30]Śniatycki, J., ‘On the geometric structure of classical field theory in Lagrangian formulation’, Proc. Camb. Phil. Soc. 68 (1970), 475–484.CrossRefGoogle Scholar
[31]Swann, A., ‘Hyper-Kähler and quaternionic Kähler geometry’, Math. Ann. 289 (1991), 421–450.CrossRefGoogle Scholar
[32]Thompson, G., ‘Symplectic manifolds with Lagrangian fibration’, Lett. Math. Phys. 12 (1986), 241–248.CrossRefGoogle Scholar
[33]Tulczyjew, W. M., ‘Homogeneous symplectic formulation of field dynamics and the PoincaréCartan form’, in: Differential geometrical methods in mathematical physics, Lecture Notes in Math. 836 (Springer, Berlin, 1980) pp. 484–497.CrossRefGoogle Scholar
[34]Weinstein, A., ‘Symplectic manifolds and their Lagrangian submanifolds’, Adv. Math. 6 (1971), 329–346.CrossRefGoogle Scholar
[35]Westwick, R., ‘Irreducible lengths of trivectors of rank seven and eight’, Pacific J. Math. 80 (1979), 575–579.CrossRefGoogle Scholar
[36]Westwick, R., ‘Real trivectors of rank seven’, Linear and Multilinear Algebra 10 (1981), 183–204.CrossRefGoogle Scholar
[37]Yano, K. and Ishihara, S., Tangent and cotangent bundles (Marcel Dekker, New York, 1973).Google Scholar