Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△b the s-...Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△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 〉 4＋2√3 ≈ 7.46 or s 〈 4- 2√3≈ 0.54 and s ≠ 0, then g is Kahler. We also show that, when n = 2, g is always Kahler unless s=2. Therefore non-Kahler compact Gauduchon fiat surfaces are exactly isosceles Hopf surfaces.展开更多
On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of...On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).展开更多
文摘Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△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 〉 4＋2√3 ≈ 7.46 or s 〈 4- 2√3≈ 0.54 and s ≠ 0, then g is Kahler. We also show that, when n = 2, g is always Kahler unless s=2. Therefore non-Kahler compact Gauduchon fiat surfaces are exactly isosceles Hopf surfaces.
基金supported by National Natural Science Foundation of China(Grant Nos.10831008,11025103 and 11501505)。
文摘On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).
