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.