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Some rigid results on doubly twisted product complex Finsler manifolds

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Science China Mathematics 2024年

作者： Hongchuan Xia Wei Xiao chunping zhong School of Mathematics and Statistics Xinyang Normal University School of Mathematical Sciences Xiamen University

Let(M1×（λ1,λ2）M2, F) be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds（M1, F1） and（M2, F2）. Denote by M the product manifold of M1and M2. In this paper, we first give the characterization for（M, F） to be of constant holomorphic curvature, and prove that（M, F） is of constant holomorphic curvature if and only if（M, F）,（M1, F1）, and（M2, F2） all have vanishing holomorphic curvature under the condition that ln λ1and ln λ2are both pluriharmonic ***, we obtain the classification for（M, F） to be a complex Landsberg manifold. Finally, we prove that a strongly convex doubly twisted product complex Finsler manifold（M, F） is projectively flat（resp. dually flat）if and only if λ1F1and λ2F2are both complex Minkowski metrics.

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Strongly convex weakly complex Berwald metrics and real Landsberg metrics

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Science China Mathematics 2018年第3期61卷 535-544页

作者： Yong He chunping zhong School of Mathematical Sciences Xiamen University School of Mathematical Sciences Xinjiang Normal University

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg *** result together with zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.

关键词： strongly convex Kahler Finsler metric complex Berwald metric real Landsberg metric real Berwald metric

Weakly complex Einstein-Finsler vector bundle Dedicated to Professor Tongde zhong for His 90th Birthday

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Science China Mathematics 2018年第6期61卷 1079-1088页

作者： Liling Sun chunping zhong College of Mathematics Taiyuan University of Technology Taiyuan 030024 China School of Mathematical Sciences Xiamen University Xiamen 361005 China

Let （E, F） be a complex Finsler vector bundle over a compact Kahler manifold （M, g） with Kahler form φ We prove that if （E, F） is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou （1997）, then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle （E, F） over a compact Kahler manifold （M, g） is necessarily φ-semistable and （E,F）=（E1,F1）……（Ek,Fk）,where Fj ：= F｜Ej, and each （Ej, Fj） is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a φ-stable Einstein-Hermitian vector bundle with the same factor c as （E, F）.

关键词： complex Finsler vector bundle complex Einstein-Finsler vector bundle Chern-Finsler connection mean curvature

A class of metrics and foliations on tangent bundle of Finsler manifolds

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Frontiers of Mathematics in China 2017年第2期12卷 417-439页

作者： Hongchuan XIA chunping zhong College of Mathematics and Statistics Xinyang Normal University Xinyang 464000 China School of Mathematical Sciences Xiamen University Xiamen 361005 China

Let （M, F） be a Finsler manifold, and let TMo be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of （TMo, G） and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.

关键词： Finsler manifold foliation constant flag curvature Vaismanconnection

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

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Science China Mathematics 2022年第8期65卷 1661-1678页

作者： Jun Nie chunping zhong School of Mathematical Sciences Xiamen UniversityXiamen 361005China

Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from *** that N is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative *** this paper,we establish a Schwarz lemma for holomorphic mappings f from M into *** applications,we obtain a Liouville type rigidity result for holomorphic mappings f from M into N,as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.

关键词： Schwarz lemma Kähler manifold complex Finsler manifold holomorphic sectional curvature

On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics

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Chinese Annals of Mathematics,Series B 2016年第2期37卷 161-174页

作者： Hongchuan XIA chunping zhong School of Mathematical Sciences Xiamen University Xiamen 361005 Fujian China

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M  C^n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S^(2n-1) M are great circles.

关键词： Complex Finsler metrics Weakly complex Berwald metrics Closed geodesics

On U(n)-invariant strongly convex complex Finsler metrics

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Science China Mathematics 2021年第11期64卷 2461-2478页

作者： Kun Wang Hongchuan Xia chunping zhong School of Mathematical Sciences Xiamen UniversityXiamen 361005China College of Mathematics and Statistics Xinyang Normal UniversityXinyang 464000China

In this paper,we obtain a necessary and sufficient condition for a U(n)-invariant complex Finsler metric F on domains in C^(n) to be strongly convex,which also makes it possible to investigate the relationship between real and complex Finsler geometries via concrete and computable *** prove a rigid theorem which states that a U(n)-invariant strongly convex complex Finsler metric F is a real Berwald metric if and only if F comes from a U(n)-invariant Hermitian *** give a characterization of U(n)-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of the U(n)-invariant strongly pseudoconvex complex Finsler ***,we prove that the real geodesics of some U(n)-invariant complex Finsler metric restricted on the unit sphere S^(2n-1)■C^(n) share a specific property as that of the complex Wrona metric on C^(n).

关键词： U(n)-invariant complex Finsler metric strongly convex real Berwald metric holomorphic curvature

A REMARK ON GENERAL COMPLEX(α,β)METRICS

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Acta Mathematica Scientia 2021年第3期41卷 670-678页

作者： Hongchuan XIA chunping zhong School of Mathematics and Statistics Xinyang Normal UniversityXinyang 464000China School of Mathematical Sciences Xiamen UniversityXiamen 361005China

In this paper,we give a characterization for the general complex(α,β)metrics to be strongly *** an application,we show that the well-known complex Randers metrics are strongly convex complex Finsler metrics,whereas ... 详细信息

关键词： Strongly convex strongly pseudoconvex general complex(α β)metrics

Foliations on the tangent bundle of Finsler manifolds

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Science China Mathematics 2012年第3期55卷 647-662页

作者： PEYGHAN Esmaei TAYEBI Akbar zhong chunping Department of Mathematics Faculty of Science Arak University Arak *** Iran Department of Mathematics Faculty of Science Qom University Qom *** Iran School of Mathematical Sciences Xiamen University Xiamen 361005 China

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by *** this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).

关键词： Finsler manifold g-natural metrics Riemannian foliation