Harmonic 1-Forms on Compact f-Manifolds

Abstract

We consider a Riemannian manifold with a compatible f -structure which admits a parallelizable kernel. With some additional integrability conditions it is called S-manifold. This class of manifolds is a natural generalization of the Sasakian manifolds. We study properties of harmonic 1-forms on such a manifold and deduce some topological properties.

HARMONIC AND HOLOMORPHIC VECTOR FIELDS ON AN Ƒ-Manifold WITH PARALLELIZABLE KERNEL

“…Then we can adapt some results on harmonic 1-forms proved in [9] to harmonic vector fields on a compact S-manifold (M, φ, ξ i , η i , g), i ∈ {1, . .…”

“…, s} we have R ♯ (ξ i ) = 2nξ. Again from [9], we know that the space H 1 of the harmonic 1-forms on a compact S-manifold M orthogonally decomposes as H 1…”

“…The third section is dedicated to the study of the harmonic vector fields on a compact S-manifold M 2n+s while in the fourth section we present some examples of harmonic 1-forms or vector fields on particular compact S-manifolds, obtained using some results of [9]. In the fifth section from certain properties of the harmonic 1-forms we get that the Ricci curvature assumes strictly positive and strictly negative values.…”