In this paper, we give a complete conformal classification of the regular space-like hyper- surfaces in the de Sitter Space S~＋1 with parallel para-Blaschke tensors.
Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of ...Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.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
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
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
Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constan...Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).展开更多
The most important Moebius invariants in the Moebius differential geometry of submanifolds in S^n＋p are the Moebius metric g, the Moebius second fundamental form B, the Moebius form φ and the Blaschke tensor A. In t...The most important Moebius invariants in the Moebius differential geometry of submanifolds in S^n＋p are the Moebius metric g, the Moebius second fundamental form B, the Moebius form φ and the Blaschke tensor A. In this paper, we obtain the upper bound of the Moebius scalar curvature of submanifolds with parallel Moebius form in S^n＋p.展开更多
基金Supported by Foundation of Natural Sciences of China(Grant Nos.11671121,11171091 and 11371018)
文摘In this paper, we give a complete conformal classification of the regular space-like hyper- surfaces in the de Sitter Space S~＋1 with parallel para-Blaschke tensors.
基金Project supported by the Stress Supporting Subject Foundation of Zhejiang Province, China
文摘Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.
基金the National Natural Science Foundation of China ( 1 0 2 71 1 0 6) ,the Natural ScienceFoundation of Zhejiang Province ( M1 0 30 4 7) and the Education Department of Zhejiang Province( 2 0 0 30 342 )
文摘In this paper,the rigidity theorems of the submanifolds in S n+p with parallel Mobius form and constant Mobius scalar curvature are given.
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.
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).
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.
基金supported by National Natural Science Foundation of China(Grant No.11531012)China Postdoctoral Science Foundation(Grant No.BX20180274)Natural Science Foundation of Zhejiang Province(Grant No.LY20A010024)。
文摘Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).
基金Supported by the NSF of China(10671087)Supported by the NSF of Jiangxi Province(2008GZS0024)
文摘The most important Moebius invariants in the Moebius differential geometry of submanifolds in S^n＋p are the Moebius metric g, the Moebius second fundamental form B, the Moebius form φ and the Blaschke tensor A. In this paper, we obtain the upper bound of the Moebius scalar curvature of submanifolds with parallel Moebius form in S^n＋p.
