In this paper,we studyλ-biharmonic hypersurfaces M_(r)^(5) of 6-dimensional pseudo Riemannian space form N_(p)^(6)(c)with the indexs 0≤p≤6,r=p−1 or p,and constant curvature c.It was proved that if the shape operato...In this paper,we studyλ-biharmonic hypersurfaces M_(r)^(5) of 6-dimensional pseudo Riemannian space form N_(p)^(6)(c)with the indexs 0≤p≤6,r=p−1 or p,and constant curvature c.It was proved that if the shape operator of M_(r)^(5) is diagonalizable,then the mean curvature is a constant.As an application,we find some types of biharmonic hypersurfaces of N_(p)^(6)(c)are minimal.展开更多
For submanifolds in a cosymplectic space form tangent to the structure vector field ξ, two important inequalities with Ricci curvature, scalar curvature and the squared mean curvature are obtained. These results are ...For submanifolds in a cosymplectic space form tangent to the structure vector field ξ, two important inequalities with Ricci curvature, scalar curvature and the squared mean curvature are obtained. These results are also applied to get corresponding consequences for anti-invariant submanifolds.展开更多
We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck form...We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L^p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-Einstein manifolds.Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact K?hler manifolds and noncompact K?hler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge,these kinds of results have not been reported.展开更多
In this paper, we obtain Chen＇s inequalities for totally real submanifolds in complex space forms with a semi-symmetric metric connection. Also, some results of A. Mihai and C. Ozgiir＇s paper have been modified.
By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42(1):165-178],we establish some inequalities for...By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42(1):165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.展开更多
A submanifold in a complex space form is called slant if it has constant Wirtinger angles. B. Y. Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP2 and CH2. So it seems that the sla...A submanifold in a complex space form is called slant if it has constant Wirtinger angles. B. Y. Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP2 and CH2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen’s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersions in CPn and CHn with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen’s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.展开更多
Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compac...Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.展开更多
Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then un...Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.展开更多
In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the...In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.展开更多
In this paper,we study the stability of the domains in a surface of constant meam curvaturein space form N3(c)by means of the estimation of the Gaussian curvature of comformal metric onthis surface.
In this paper,we prove that four-dimensional hypersurface M_(r)^(4) with proper mean curvature vector field(i.e.,ΔHis proportional toH)in pseudo-Riemannian space form N_(s)^(5)(c) has constant mean curvature,and give...In this paper,we prove that four-dimensional hypersurface M_(r)^(4) with proper mean curvature vector field(i.e.,ΔHis proportional toH)in pseudo-Riemannian space form N_(s)^(5)(c) has constant mean curvature,and give the value or range of this constant.As an application,we obtain that biharmonic hypersurfaces in N_(s)^(5)(c) are minimal in some specific case.展开更多
In this paper, we establish some formulas on closed curves in 2-dimensionalspace forms. Mean absolute geodesic curvature is introduced to describe the average curving of aclosed curve. In this sense, a closed curve co...In this paper, we establish some formulas on closed curves in 2-dimensionalspace forms. Mean absolute geodesic curvature is introduced to describe the average curving of aclosed curve. In this sense, a closed curve could be compared with a geodesic circle that is theboundary of a convex geodesic circular disk containing the closed curve. The comparison can be usedto show some properties of space forms only on themselves.展开更多
Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to ...Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.展开更多
Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n ...Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n dimensional hyperbolic cylinder in the (2n-1) dimensional pseudo hyperbolic space.展开更多
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersu...In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.展开更多
Superprocess is one class of measure-valued branching Markov processes. Many results of the process on abstract spaces and Euclidean spaces are obtained in the literature. In this paper, we discuss super-Brownian moti...Superprocess is one class of measure-valued branching Markov processes. Many results of the process on abstract spaces and Euclidean spaces are obtained in the literature. In this paper, we discuss super-Brownian motions on two space forms and reveal partially the relationship between the properties of superprocesses and the geometric structure of the underlying state spaces. Finally, we also presents some open problems.展开更多
We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those t...We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one.In the case where the ambient space is Cn,the quasi umbilical case was studied in Blair(2007).However,the classification there is not complete and several examples are missing.Here,we complete(and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.展开更多
By using Darboux transformations, the authors give the explicit construction for local iso-metric immersions of space forms Mn(c) into space forms M2n-1(c + ε2) via purely algebraicalgorithm.
In this paper, we investigate the space of L<sup>p</sup> p-harmonic 1-forms on a complete noncompact orientable δ-stable hypersurface M<sup>m</sup> that is immersed in space form <img src=&...In this paper, we investigate the space of L<sup>p</sup> p-harmonic 1-forms on a complete noncompact orientable δ-stable hypersurface M<sup>m</sup> that is immersed in space form <img src="Edit_6fbc11b9-ac23-40e2-b045-0fb25419337d.png" width="35" height="23" alt="" /> with nonnegative BiRic curvature. We prove the nonexistence of L<sup>p</sup> p-harmonic 1-forms on M<sup>m</sup>. Moreover, we obtain some vanishing properties for this class of harmonic 1-forms.展开更多
基金Supported by National Natural Science Foundation of China(12161078)Foundation for Innovative Fundamental Research Group Project of Gansu Province(24JRRA778)Project of Northwest Normal University(20240010)。
文摘In this paper,we studyλ-biharmonic hypersurfaces M_(r)^(5) of 6-dimensional pseudo Riemannian space form N_(p)^(6)(c)with the indexs 0≤p≤6,r=p−1 or p,and constant curvature c.It was proved that if the shape operator of M_(r)^(5) is diagonalizable,then the mean curvature is a constant.As an application,we find some types of biharmonic hypersurfaces of N_(p)^(6)(c)are minimal.
基金Supported by the Natural Science Foundation of Henan(004051900)
文摘For submanifolds in a cosymplectic space form tangent to the structure vector field ξ, two important inequalities with Ricci curvature, scalar curvature and the squared mean curvature are obtained. These results are also applied to get corresponding consequences for anti-invariant submanifolds.
基金supported by the Foundation for training Young Teachers in University of Shanghai(ZZegd16003)supported by National Natural Science Foundation of China(11271071,11771087)LMNS,Fudan University
文摘We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L^p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-Einstein manifolds.Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact K?hler manifolds and noncompact K?hler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge,these kinds of results have not been reported.
基金the Foundation for Excellent Young Talents of Higher Education of Anhui Province(GrantNo.2011SQRL021ZD)
文摘In this paper, we obtain Chen＇s inequalities for totally real submanifolds in complex space forms with a semi-symmetric metric connection. Also, some results of A. Mihai and C. Ozgiir＇s paper have been modified.
基金Supported by the NSFC of China(11001076)Supported by the NSF of Henan Provincial Education Department(2010A110008)
文摘By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42(1):165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.
基金This project is supported by the NSFC(10271041)Tianyuan Youth Foundation of Mathematics.
文摘A submanifold in a complex space form is called slant if it has constant Wirtinger angles. B. Y. Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP2 and CH2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen’s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersions in CPn and CHn with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen’s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.
文摘Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.
文摘Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.
文摘In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.
基金Supported by N.NSF of China(19871038)Supported by the NSF of Jiangxi(981105)
文摘In this paper,we study the stability of the domains in a surface of constant meam curvaturein space form N3(c)by means of the estimation of the Gaussian curvature of comformal metric onthis surface.
基金Supported by the National Natural Science Foundation of China(Grant No.12161078)the Foundation for Innovative Fundamental Research Group Project of Gansu Province(Grant No.24JRRA778)。
文摘In this paper,we prove that four-dimensional hypersurface M_(r)^(4) with proper mean curvature vector field(i.e.,ΔHis proportional toH)in pseudo-Riemannian space form N_(s)^(5)(c) has constant mean curvature,and give the value or range of this constant.As an application,we obtain that biharmonic hypersurfaces in N_(s)^(5)(c) are minimal in some specific case.
文摘In this paper, we establish some formulas on closed curves in 2-dimensionalspace forms. Mean absolute geodesic curvature is introduced to describe the average curving of aclosed curve. In this sense, a closed curve could be compared with a geodesic circle that is theboundary of a convex geodesic circular disk containing the closed curve. The comparison can be usedto show some properties of space forms only on themselves.
基金supported by NSFC(Grant Nos.12371195,12022111)the Fundamental Research Funds for the Central Universities(Grant No.2042023kf0207)+1 种基金the second author was partially supported by NSFC(Grant No.11831009)Fundings of Innovating Activities in Science and Technology of Hubei Province。
文摘Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.
基金the National Natural Science Foundationof China (No.1970 10 17
文摘Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n dimensional hyperbolic cylinder in the (2n-1) dimensional pseudo hyperbolic space.
基金supported by National Natural Science Foundation of China (Grant Nos. 11971253 and 11471246)AnHui Natural Science Foundation (Grant No. 1608085MA03)。
文摘In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.
文摘Superprocess is one class of measure-valued branching Markov processes. Many results of the process on abstract spaces and Euclidean spaces are obtained in the literature. In this paper, we discuss super-Brownian motions on two space forms and reveal partially the relationship between the properties of superprocesses and the geometric structure of the underlying state spaces. Finally, we also presents some open problems.
基金supported by the Ministry of Education,Science and Technological Development of the Republic of Serbia(Grant No.174012)。
文摘We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one.In the case where the ambient space is Cn,the quasi umbilical case was studied in Blair(2007).However,the classification there is not complete and several examples are missing.Here,we complete(and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.
基金Project supported by the National Natural Science Foundation of China (No.10271106)the Science Foundation of the Ministry of Education of Chinathe Natural Science Foundation of Zhejiang Province, China.
文摘By using Darboux transformations, the authors give the explicit construction for local iso-metric immersions of space forms Mn(c) into space forms M2n-1(c + ε2) via purely algebraicalgorithm.
文摘In this paper, we investigate the space of L<sup>p</sup> p-harmonic 1-forms on a complete noncompact orientable δ-stable hypersurface M<sup>m</sup> that is immersed in space form <img src="Edit_6fbc11b9-ac23-40e2-b045-0fb25419337d.png" width="35" height="23" alt="" /> with nonnegative BiRic curvature. We prove the nonexistence of L<sup>p</sup> p-harmonic 1-forms on M<sup>m</sup>. Moreover, we obtain some vanishing properties for this class of harmonic 1-forms.
