The Wu-Yau theorem on Sasakian manifolds

Abstract

In this note, We proved that a compact Sasakian manifold $(M,\xi ,\eta ,\Phi ,g)$ with negative transverse holomorphic sectional curvature must have has a Sasakian structure $(\xi ,\eta ,\Phi ,g)$ with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with non-positive transverse holomorphic sectional curvature, then the negative first basic Chern class is transverse nef and we have the Miyaoka-Yau type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.

18 pages