论文标题
在等距沉浸式近乎$ k $ - 产品歧管上
On isometric immersions of almost $k$-product manifolds
论文作者
论文摘要
带有$ k \ ge2 $互补的成对正交分布的riemannian流形被称为riemannian几乎$ k $ - 产品歧管。在文章中，我们首次研究了以下问题：找到固有的和外在的不变性之间的关系，几乎是$ k $ - 产品 - 产品歧管等于浸入另一种riemannian歧管中。对于这种沉浸式，我们建立了最佳的不平等，其中包括混合标态曲率和平均曲率的平方。尽管Riemannian曲率张量属于固有的几何形状，但一个称为混合曲率的特殊部分也与Riemannian几乎$ K $ - 产品歧管的外部几何形状有关。我们的不平等还包含与B.-Y Chen的$δ$ -Invariants相关的混合标态曲率类型。给出了用于多重扭曲和翘曲产品的等距沉浸式的应用（我们通过用我们的不变剂代替截面曲率来改善一些已知的最佳不平等现象），以及非浸泡和不存在的叶层亚法叶子的不存在的问题。
A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost $k$-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's $δ$-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known optimal inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds.
