Lagrangian submanifolds satisfying a basic equality

被引:20
作者
Chen, BY [1 ]
Vrancken, L [1 ]
机构
[1] DEPT WISKUNDE, B-3001 LOUVAIN, BELGIUM
关键词
D O I
10.1017/S0305004100074867
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form M(n)(4c) (c = +/-1), the squared mean curvature and the scalar curvature of M satisfy the folio-wing inequality: H-2 greater than or equal to 2(n+2)/n(2)(n-1)tau-(n+2/n)c. He then introduced three families of Riemannian n-manifolds P-a(n)(a > 1), C-a(n)(a > 1), D-a(n)(0 < a < 1) and two exceptional n-spaces F-n, L(n) and proved the existence of a Lagrangian isometric immersion p(a), from P-a(n) into CPn(4) and the existence of Lagrangian isometric immersions f, l, c(a), d(a), from F-n, L(n), C-a(n), D-a(n) into CHn(-4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesic ones, these are the only Lagrangian submanifolds in CPn(4) and in CHn(-4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into CPn(4) and CHn(-4).
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页码:291 / 307
页数:17
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