On the Global Geometry of Harmonic Symmetric Bilinear Differential Forms

A harmonic symmetric $p$-form $\varphi$ is defined as an element of the kernel of a self-adjoint differential operator $\square$. By using the properties of this operator, the dimension of the $\mathbb R$-modulus of harmonic symmetric $p$-forms is shown to be finite on a compact Riemannian manifold. A nonexistence theorem is proved for harmonic symmetric $2$-forms tangent to the boundary of a compact Riemannian manifold.