We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics o...We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S^3 with Ric = 2F^2, Ric = 0 and Ric =-2F^2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.展开更多
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represent...This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi- Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are K/ihler Calabi-Yau surfaces and Hopf surfaces.展开更多
We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat -manifolds and also a construction of global canonical family of holomorphic （n, 0）-forms on the...We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat -manifolds and also a construction of global canonical family of holomorphic （n, 0）-forms on the deformation spaces of the Ricci-flat -manifolds.展开更多
On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly buil...On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly built up of 4 Harvey-Lawson's models in 4directions.展开更多
Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five.In this paper,we characterize all Ricci-flat graphs of girth four ...Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five.In this paper,we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.展开更多
We give a differential-geometric construction of Calabi-Yau fourfolds by the‘doubling’method,which was introduced in Doi and Yotsutani(N Y J Math 20:1203-1235,2014)to construct Calabi-Yau threefolds.We also give exa...We give a differential-geometric construction of Calabi-Yau fourfolds by the‘doubling’method,which was introduced in Doi and Yotsutani(N Y J Math 20:1203-1235,2014)to construct Calabi-Yau threefolds.We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds.Ingredients in our construction are admissible pairs,which were first dealt with by Kovalev(J Reine Angew Math 565:125-160,2003).Here in this paper an admissible pair(X,D)consists of a compact Kähler manifold X and a smooth anticanonical divisor D on X.If two admissible pairs(X_(1),D_(1))and(X_(2),D_(2))with dimC X_(i)=4 satisfy the gluing condition,we can glue X_(1)\D_(1)and X_(2)\D_(2)together to obtain a compact Riemannian 8-manifold(M,g)whose holonomy group Hol(g)is contained in Spin(7).Furthermore,if theA-genus of M equals 2,then M is a Calabi-Yau fourfold,i.e.,a compact Ricci-flat Kähler fourfold with holonomy SU(4).In particular,if(X_(1),D_(1))and(X_(2),D_(2))are identical to an admissible pair(X,D),then the gluing condition holds automatically,so that we obtain a compact Riemannian 8-manifold M with holonomy contained in Spin(7).Moreover,we show that if the admissible pair is obtained from any of the toric Fano fourfolds,then the resulting manifold M is a Calabi-Yau fourfold by computing^A(M)=2.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11371386)the European Union’s Seventh Framework Programme (FP7/2007–2013) (Grant No. 317721)National Science Foundation of USA (Grant No. DMS-0810159)
文摘We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S^3 with Ric = 2F^2, Ric = 0 and Ric =-2F^2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
基金supported in part by NSFC(Grant No.11531012),NSFC(Grant No.11688101)supported in part by China’s Recruitment Program
文摘This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi- Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are K/ihler Calabi-Yau surfaces and Hopf surfaces.
文摘We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat -manifolds and also a construction of global canonical family of holomorphic （n, 0）-forms on the deformation spaces of the Ricci-flat -manifolds.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10101004).
文摘On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly built up of 4 Harvey-Lawson's models in 4directions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11601093,12025109,12071489 and 61976104)the Research Fund of Guangdong University of Foreign Studies(Grant Nos.299-X5219228 and 297-ZW200011)。
文摘Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five.In this paper,we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.
文摘We give a differential-geometric construction of Calabi-Yau fourfolds by the‘doubling’method,which was introduced in Doi and Yotsutani(N Y J Math 20:1203-1235,2014)to construct Calabi-Yau threefolds.We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds.Ingredients in our construction are admissible pairs,which were first dealt with by Kovalev(J Reine Angew Math 565:125-160,2003).Here in this paper an admissible pair(X,D)consists of a compact Kähler manifold X and a smooth anticanonical divisor D on X.If two admissible pairs(X_(1),D_(1))and(X_(2),D_(2))with dimC X_(i)=4 satisfy the gluing condition,we can glue X_(1)\D_(1)and X_(2)\D_(2)together to obtain a compact Riemannian 8-manifold(M,g)whose holonomy group Hol(g)is contained in Spin(7).Furthermore,if theA-genus of M equals 2,then M is a Calabi-Yau fourfold,i.e.,a compact Ricci-flat Kähler fourfold with holonomy SU(4).In particular,if(X_(1),D_(1))and(X_(2),D_(2))are identical to an admissible pair(X,D),then the gluing condition holds automatically,so that we obtain a compact Riemannian 8-manifold M with holonomy contained in Spin(7).Moreover,we show that if the admissible pair is obtained from any of the toric Fano fourfolds,then the resulting manifold M is a Calabi-Yau fourfold by computing^A(M)=2.
