The Wu-Yau theorem on Sasakian manifolds

In this paper, we proved that a compact Sasakian manifold (M,xi,eta,Phi,g) with negative transverse holomorphic sectional curvature must have a Sasakian structure (xi,eta',Phi',g') with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with nonpositive transverse holomorphic sectional curvature, then the negative first basic Chem class -c(1)(B) (M, F-xi) is transverse nef and we have the Miyaoica-Yau-type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.