论文标题
复杂双曲线二次的拉格朗日亚曼叶
Lagrangian submanifolds of the complex hyperbolic quadric
论文作者
论文摘要
我们将复杂的双曲线二次$ {q^*}^n $视为复杂的抗DE保姆空间的复杂性超脸。该子手机的形状操作员在$ {q^*}^n $上产生了一个本地几乎产品结构的家族，然后将其用于在任何$ {q^*}^n $的Lagrangian Submanifold上定义局部角度函数。我们证明，拉格朗日浸入$ {q^*}^n $中可以看作是（真实）抗DE安静的空间的高空表情的高斯图，并将角度函数与此hypersurface的主要曲率相关联。我们还提供了一个公式，该公式将拉格朗日浸入这些主要曲率的平均曲率有关。这些定理用几个抗DE保姆空间及其高斯图的空隙性突出的示例进行了说明。最后，我们对一些$ {q^*}^n $的Lagrangian Submanifolds的一些家庭分类为：那些具有平行第二基本形式的家族，而诱导的截面曲率是恒定的。在这两种情况下，拉格朗日亚曼菲德都被迫完全大地测量。
We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used to define local angle functions on any Lagrangian submanifold of ${Q^*}^n$. We prove that a Lagrangian immersion into ${Q^*}^n$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagrangian immersion to these principal curvatures. The theorems are illustrated with several examples of spacelike hypersurfaces of anti-de Sitter space and their Gauss maps. Finally, we classify some families of minimal Lagrangian submanifolds of ${Q^*}^n$: those with parallel second fundamental form and those for which the induced sectional curvature is constant. In both cases, the Lagrangian submanifold is forced to be totally geodesic.
