An invitation to multisymplectic geometry

Leonid Ryvkin (1) , Tilmann Wurzbacher (2)

1 Fakultät für Mathematik [Bochum]
2 IECL - Institut Élie Cartan de Lorraine

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in "multisymplectic terms". Furthermore, we investigate basic questions on normal forms of multisymplectic man-ifolds, notably the questions wether and when Darboux-type theorems hold, and "how many" diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.

Mots clés: multisymplectic geometry; Lagrangian and Hamiltonian field theories; Darboux-type theorems; groups of multisymplectic diffeomorphisms; homotopy comoment

Domaines: Géométrie différentielle [math.DG]

Leonid Ryvkin, Tilmann Wurzbacher. An invitation to multisymplectic geometry. Journal of Geometry and Physics, 2019, ⟨10.1016/j.geomphys.2019.03.006⟩. ⟨hal-02471860⟩