Analytic fields on compact balanced Hermitian manifolds
Abstract
Description
On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harmonic. We prove that if $H$ is positive definite then the first Betti number $b_1 = 0$ and the Hodge number $h^{1,0} = 0$. We obtain an obstruction to the existence of Killing vector fields in terms of the Ricci tensor of the Chern connection: if the Chern form of the Chern connection on a compact balanced Hermitian manifold is non- positive definite then every Killing vector field is analytic; if moreover the Chern form is negative definite then there are no Killing vector fields. It is proved that on a compact balanced Hermitian manifold every affine with respect to the Chern connection vector field is an analytic vector field.
14 pages, Latex format, no figures
14 pages, Latex format, no figures