On Compact Hermitian Manifolds with Flat Gauduchon Connections

Abstract

Given a Hermitian manifold (Mn, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $${\nabla ^s} = \left( {1 - \frac{s}{2}} \right){\nabla ^c} + \frac{s}{2}{\nabla ^b}$$ the s-Gauduchon connection of M, where ∇c and ∇b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 (a flat Chern connection) or s = 2 (a flat Bismut connection) are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either $$s \geqslant 4 + 2\sqrt 3 \approx 7.46$$ or $$s \leqslant 4 - 2\sqrt 3 \approx 0.54$$ and s ≠ 0, then g is Kähler. We also show that, when n = 2, g is always Kähler unless s = 2. Therefore non-Kähler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.