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.
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In Memory of Professor Qikeng Lu (1927–2015)
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Yang, B., Zheng, F.Y. On Compact Hermitian Manifolds with Flat Gauduchon Connections. Acta. Math. Sin.-English Ser. 34, 1259–1268 (2018). https://doi.org/10.1007/s10114-018-7409-y
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DOI: https://doi.org/10.1007/s10114-018-7409-y