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On the geometry of multisymplectic manifolds. (English) Zbl 0968.53052
Multisymplectic forms appear in an invariant Lagrangian (or Euler-Cartan) formulation of classical field equations. The affine dual of the first jet bundle carries canonically a closed nondegenerate $$k$$-form, that together with a Lagrangian $$L$$ gives an invariant expression for the Euler-Lagrange equations. In that respect multisymplectic forms can be seen as generalization of symplectic forms. There are however other generalizations: The linear (not affine) dual of the first jet bundle carries naturally a closed nondegenerate vector valued two form, called polysymplectic form that leads directly to the Hamiltonian form of field equations. The article under review studies only basic properties of multisymplectic forms: The first half deals with the linear case, with examples of forms on exterior product spaces, Lagrangian subspaces, and linear canonical forms. The remainder of the article presents multisymplectic manifolds: examples, canonical forms on the exterior bundle and Lagrangian fibrations. No jet bundle structures or field equations are studied.
Reviewer: C.Günther (Libby)

##### MSC:
 53D05 Symplectic manifolds (general theory) 58A20 Jets in global analysis