Lagrangian isometric immersions of a real-space-form Mn(c) into a complex-space-form M˜n(4c)
Lagrangian isometric immersions of a real-space-form Mn(c) into a complex-space-form M˜n(4c)
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DOI：
10.1017/s030500419700217x
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发表时间：
1998-07
影响因子：
0.8
通讯作者：
Bang‐Yen Chen;F. Dillen;L. Verstraelen;L. Vrancken
中科院分区：
文献类型：
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作者：
Bang‐Yen Chen;F. Dillen;L. Verstraelen;L. Vrancken
众所周知，具有常全纯截面曲率4c的复空间形式M n（4c）的全测地拉格朗日子流形是具有常截面曲率c的实空间形式。本文研究并确定了常截面曲率c的实空间形式到复空间形式M_n（4c）的非全测地拉格朗日等距浸入。为了做到这一点，与形式为f1 I1 ×... ×fkIk× 1 Nn −k（c）的实空间形式的每一个扭积分解相关联，我们引入一个规范1-形式，称为扭积分解的扭量形式。粗略地说，我们的主要结果说，如果扭量形式的这样一个扭曲的产品分解的简单连接的实空间形式的恒定截面曲率c是扭曲封闭的，那么它允许一个“唯一的”适应拉格朗日等距浸入到一个复杂的空间形式M n（4c）。相反，如果L：Mn（c）→ M n（4c）是具有常截面曲率c的实空间形式Mn（c）到复空间形式M n（4c）的非全测地拉格朗日等距浸入，则Mn（c）允许一个适当的扭积分解，具有扭闭扭量形式，并且拉格朗日浸入L由扭积的相应的适应拉格朗日等距浸入给出.在本文中，我们还提供了显式结构的适应拉格朗日等距浸入的一些自然扭曲的产品分解的实空间形式。
It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.