Weyl structures with positive Ricci tensor

Abstract

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number $b_1=1$ and Hodge numbers $h^{p,0}=0$ for $p>0$, $h^{0,1}=1$, $h^{0,q}=0$ for $q>1$.

8 pages, Latex format, no figures; added section; to appear in Diff. Geom. Appl