Analytic fields on compact balanced Hermitian manifolds

Details Analytic fields on compact balanced Hermitian manifolds Analytic fields on compact balanced Hermitian manifolds

1996-06-24

arxiv.org/abs/dg-ga/9606011v1

Mathematics

Differential Geometry

14 pages, Latex format, no figures

Scientific paper

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.

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