$$L^2$$-transverse Killing forms on a transverse Kähler foliation

Abstract

It is well-known that on a compact Riemannian manifold with a transverse Kahler foliation of codimension $$q=2m$$, any transverse Killing r-form $$(r\ge 2)$$ is parallel (Jung and Jung in Bull Korean Math Soc 49:445–454, 2012). In this article, we study the parallelness of $$L^2$$-transverse Killing forms on a complete foliated Riemannian manifold M with a transverse Kahler foliation $${\mathcal {F}}$$. Precisely, if all leaves of $${\mathcal {F}}$$ are compact and the mean curvature form is transversally holomorphic, coclosed and bounded, then all $$L^2$$-transverse Killing r-forms $$(r\ge 2)$$ are parallel. In addition, if the volume of M is infinite, then all $$L^2$$-transverse Killing r-forms $$(r\ge 2)$$ are trivial.