We obtain an inequality in Sm×R and Hm×R which is similar to DDVV conjecture.As an application,we show that a minimal submanifold in H m×R with nonnegative scalar curvature must be a surface of the type...We obtain an inequality in Sm×R and Hm×R which is similar to DDVV conjecture.As an application,we show that a minimal submanifold in H m×R with nonnegative scalar curvature must be a surface of the type γ×R,where γ is a geodesic in H m.In addition,we get a pinching theorem in Sm×R.展开更多
Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal su...Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of MSbius geometry. We classify Wintgen ideal submanfiolds of dimension rn ≥ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution D2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if D2 generates a k-dimensional integrable distribution Dk and k 〈 m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.展开更多
Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This pro...Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This property is conformal invariant;hence we study them in the framework of Mbius geometry,and restrict to three-dimensional Wintgen ideal submanifolds in S5.In particular,we give Mbius characterizations for minimal ones among them,which are also known as(3-dimensional)austere submanifolds(in 5-dimensional space forms).展开更多
基金supported by National Natural Science Foundation of China (Grant No.10871149)Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200804860046)
文摘We obtain an inequality in Sm×R and Hm×R which is similar to DDVV conjecture.As an application,we show that a minimal submanifold in H m×R with nonnegative scalar curvature must be a surface of the type γ×R,where γ is a geodesic in H m.In addition,we get a pinching theorem in Sm×R.
基金The authors thank Dr. Zhenxiao Xie for helpful discussion which deepen the understanding of the main results, and they are grateful to the referees for their critical viewpoints and suggestions, which improve the exposition and correct many errors. This work was supported by the National Natural Science Foundation of China （Grant No. 11171004） Xiang Ma was partially supported by the National Natural Science Foundation of China （Grant No. 10901006） and Changping Wang was partially supported by the National Natural Science Foundation of China （Grant No. 11331002）.
文摘Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of MSbius geometry. We classify Wintgen ideal submanfiolds of dimension rn ≥ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution D2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if D2 generates a k-dimensional integrable distribution Dk and k 〈 m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.
基金supported by National Natural Science Foundation of China (Grant Nos. 10901006,11171004 and 11331002)
文摘Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This property is conformal invariant;hence we study them in the framework of Mbius geometry,and restrict to three-dimensional Wintgen ideal submanifolds in S5.In particular,we give Mbius characterizations for minimal ones among them,which are also known as(3-dimensional)austere submanifolds(in 5-dimensional space forms).
基金partially supported by NSFC(No.12171037)the Fundamental Research Funds for the Central Universities+5 种基金partially supported by NSFC(Nos.12171037,12271040)China Postdoctoral Science Foundation(No.2022M720261)partially supported by NSFC(Nos.11931007,11871282)Nankai Zhide Foundation and Tianjin Outstanding Talents FoundationChina Postdoctoral Science Foundation(No.BX20230018)National Key R&D Program of China(No.2020YFA0712800)。
