The Wu-Yau theorem on Sasakian manifolds

TL;DR

This paper proves that compact Sasakian manifolds with negative or non-positive transverse holomorphic sectional curvature have specific curvature properties and satisfy certain inequalities, extending classical results to the Sasakian setting.

Contribution

It establishes the Sasakian analogues of the Wu-Yau theorem, linking transverse curvature conditions to the negativity of the transverse Ricci curvature and related inequalities.

Findings

Negative transverse holomorphic sectional curvature implies negative transverse Ricci curvature.

Non-positive transverse holomorphic sectional curvature leads to the transverse nef property of the first basic Chern class.

Quasi-negative transverse holomorphic sectional curvature yields a Chern number inequality.

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.