Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume

Symmetry Integrability and Geometry-Methods and Applications

article

Kenshiro Tashiro¹ 

[1] Department of Mathematics, Tohoku University

关键词: sub-Riemannian geometry;    Carnot groups;    Popp's volume;    systole.;   

DOI  :  10.3842/SIGMA.2022.058

来源: National Academy of Science of Ukraine

【 摘 要 】

In this paper, we give a systolic inequality for a quotient space of a Carnot group $\Gamma\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $\Gamma\backslash G$ is less than ${\rm Cvol}(\Gamma\backslash G)^{\frac{1}{Q}}$, 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.

【 授权许可】

Unknown