The purpose of this paper is to give a sufficient and necessary condition of totally geodesic on invariant submanifold of contact metric manifold and is to generalize the results in [3] and [4].
In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) ...In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space P^(k).This is a generalization of Cartan’s Second Main Theorem.As a consequence,we establish a uniqueness theorem for holomorphic mappings which intersect O(k^(3)) many totally geodesic hypersurfaces.展开更多
In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic subman...In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.展开更多
In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.
Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of m...Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.展开更多
In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of ...In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of the length of the second fundamental form of a minimal subrnanifold in the unit sphere is less than 2n/3+δ(n), it must be totally geodesic or diffeomorphic to a Veronese surface.展开更多
In the present paper,the authors study totally real 2-harmonic submanifolds in a complex space form and obtain a Simons' type integral inequality of compact submanifolds as well as some relevant conclusions.
In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associate...In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.展开更多
We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian sub...We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly KahlerS3 × S3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function A, such that g（（△↓h）（v, v, v）, Jr） = λ holds for all unit tangent vector v.展开更多
Let Pt denote the tubular hypersurface of radius t around a given compatible submanifold in a symmetric space of arbitrary rank. The authors will obtain some relations between the integrated mean curvatures of P, and ...Let Pt denote the tubular hypersurface of radius t around a given compatible submanifold in a symmetric space of arbitrary rank. The authors will obtain some relations between the integrated mean curvatures of P, and their derivatives with respect to f. Moreover, the authors will emphasize the differences between the results obtained for rank one and arbitrary rank symmetric spaces.展开更多
et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, ...et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, then Mi is isometric to s^ni/Zh, CP^m/2, or CP^ni/2/Z2 with the canonical metric when ni 〉 0, and thus, M is isometric to Sn/Zh, CPn/2, or CPn/2/Z2 except possibly iso when n = 3 and M1 （or M2） ≌ S1/Zh with h ≥ 2 or n iso= 4 and M1 （or M2） iso ≌ RP^2展开更多
文摘The purpose of this paper is to give a sufficient and necessary condition of totally geodesic on invariant submanifold of contact metric manifold and is to generalize the results in [3] and [4].
基金partially supported by a graduate studentship of HKU,the RGC grant(1731115)the National Natural Science Foundation of China(11701382)partially supported by the RGC grant(1731115 and 17307420)。
文摘In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space P^(k).This is a generalization of Cartan’s Second Main Theorem.As a consequence,we establish a uniqueness theorem for holomorphic mappings which intersect O(k^(3)) many totally geodesic hypersurfaces.
基金Research partially supported by the Ministry of Science and Environmental Protectipn of Serbia, Project 1646Research partially supported by EGIDE, Pavle Savic 07945VC(France)
文摘In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.
文摘In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.
基金supported by National Natural Science Foundation of China(Grant Nos.11071248 and 11226079)Program of Natural Science Research of Jiangsu Higher Education Institutions of China(Grant No.12KJD110004)
文摘Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.
基金Supported by the National Natural Scieuce Foundation of China(19971081)
文摘In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of the length of the second fundamental form of a minimal subrnanifold in the unit sphere is less than 2n/3+δ(n), it must be totally geodesic or diffeomorphic to a Veronese surface.
基金Natural Science Foundation of Education Department of Anhui Province （No. 2004kj166zd）.
文摘In the present paper,the authors study totally real 2-harmonic submanifolds in a complex space form and obtain a Simons' type integral inequality of compact submanifolds as well as some relevant conclusions.
基金supported by National Natural Science Foundation of China (Grant No.11071248)Knowledge Innovation Funds of CAS (Grant No.KJCX3-SYW-S03)the President Fund of GUCAS
文摘In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.
基金supported by National Natural Science Foundation of China (Grant No. 11371330)
文摘We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly KahlerS3 × S3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function A, such that g（（△↓h）（v, v, v）, Jr） = λ holds for all unit tangent vector v.
文摘Let Pt denote the tubular hypersurface of radius t around a given compatible submanifold in a symmetric space of arbitrary rank. The authors will obtain some relations between the integrated mean curvatures of P, and their derivatives with respect to f. Moreover, the authors will emphasize the differences between the results obtained for rank one and arbitrary rank symmetric spaces.
文摘et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, then Mi is isometric to s^ni/Zh, CP^m/2, or CP^ni/2/Z2 with the canonical metric when ni 〉 0, and thus, M is isometric to Sn/Zh, CPn/2, or CPn/2/Z2 except possibly iso when n = 3 and M1 （or M2） ≌ S1/Zh with h ≥ 2 or n iso= 4 and M1 （or M2） iso ≌ RP^2
