论文标题
两个正交不相互分布的相互曲率的变化
Variations of the mutual curvature of two orthogonal non-complementary distributions
论文作者
论文摘要
在带有分布的平滑歧管上，$ {\ cal d} _1 $和$ {\ cal d} _2 $具有微不足道的交叉点，我们将其相互曲率的整体视为Riemannian量级的函数，使得分布正交。相互曲率定义为所有对矢量跨个性基础跨越平面的截面曲率的总和，使得对向量的一个向量属于$ {\ cal d} _1 $，第二个向量属于$ {\ cal d} _2 _2 $。因此，它在平面场的截面曲率之间进行插值（如果两个分布都是一维的），以及Riemannian几乎几乎产物结构的混合标量曲率（如果两个分布在一起均跨越切线束）。我们得出了欧拉 - 拉格朗日方程的功能，该功能是根据分布的外在几何形状（即它们的第二个基本形式和集成性张量）制定的。我们为在Riemannian订阅域，扭曲产品和$ f $ - $ K $ -CONTACT歧管的域定义的分布中提供了关键指标的示例。
On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual curvature is defined as the sum of sectional curvatures of planes spanned by all pairs of vectors from an orthonormal basis, such that one vector of the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\cal D}_2$. As such, it interpolates between the sectional curvature of a plane field (if both distributions are one-dimensional), and the mixed scalar curvature of a Riemannian almost product structure (if both distributions together span the tangent bundle). We derive Euler-Lagrange equations for the functional, formulated in terms of extrinsic geometry of distributions, i.e., their second fundamental forms and integrability tensors. We give examples of critical metrics for distributions defined on domains of Riemannian submersions, twisted products and $f$-$K$-contact manifolds.
