Carnot群的紧商与Popp体积的广义不等式
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume
摘要：
本文给出了具有Popp体积的卡诺群$Γ\backslash G$的商空间的收缩不等式.也就是说，我们证明了存在一个正的常数$C$，使得$Γ\backslash G$的收缩小于${\rm Cvol}(Γ\backslash G)^{\frac{1}{Q}}$，其中$Q$是豪斯多夫维数。此外，该常数仅依赖于李代数$\mathfrak{g}=\bigoplus V_i$的分次维数。为了证明这一事实，在Popp体积定义中引入的$G$上的标积起着关键作用。
英文摘要：
In this paper, we give a systolic inequality for a quotient space of a Carnot group $Γ\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $Γ\backslash G$ is less than ${\rm Cvol}(Γ\backslash G)^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=\bigoplus V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.
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