Density-valued symplectic forms from a multisymplectic viewpoint
Laura Leski, Leonid Ryvkin
TL;DR
The paper identifies and analyzes density-valued (k+2)-forms (dvs-forms) as a natural multisymplectic generalization of volume and symplectic structures. It provides an intrinsic normal-form criterion: when the pointwise model has constant $F(ω)$-rank and the annihilator distribution $D(ω)$ is involutive, a local Darboux-type form exists, obtained via a splitting and Moser-type flow; it further proves that $D(ω)$ is automatically involutive for most dimensions ($m\neq2$). The work also investigates global and infinitesimal symmetries of dvs-forms, showing a rich leaf-wise symmetry structure and describing how conformal symplectic behavior arises when the symplectic part is exact. Finally, it links dvs-forms to cosymplectic geometry, illustrating how cosymplectic data yield dvs-forms and how cosymplectic Darboux theory informs the multisymplectic setting.
Abstract
We give an intrinsic characterization of multisymplectic manifolds that have the linear type of density-valued symplectic forms in each tangent space, prove Darboux-type theorems for these forms, and investigate their symmetries.