The Koppelman-Leray formula on complex manifolds is obtained, and under suitable condition the continuous solution of partial derivative-equation on complex manifolds is obtained.
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
An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used ...An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chern’s theorems: "The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces." Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks.展开更多
The history of Finsler geometry is reviewed and briefly recent development in Finsler geometry and its application is completed systematically. Furthermore, an interesting open problem has been proposed in this field.
The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of th...The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C 2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s≥ 3 and a a complex parameter. We will prove in the second part of the paper that for s≥ 4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent.展开更多
文摘The Koppelman-Leray formula on complex manifolds is obtained, and under suitable condition the continuous solution of partial derivative-equation on complex manifolds is obtained.
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).
基金Project supported partially by the National Natural Science Foundation of China.
文摘An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chern’s theorems: "The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces." Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks.
文摘The history of Finsler geometry is reviewed and briefly recent development in Finsler geometry and its application is completed systematically. Furthermore, an interesting open problem has been proposed in this field.
基金supported in part by US National Science Foundation (Grant No.0801056)supported in part by National Natural Science Foundation of China (Grant No.10901123)+1 种基金Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090141120010)Ky and Yu-Fen Fan Fund from American Mathematical Society, and a research fund from Wuhan University(Grant No. 1082002)
文摘The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C 2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s≥ 3 and a a complex parameter. We will prove in the second part of the paper that for s≥ 4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent.
