|
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume
Kenshiro Tashiro Department of Mathematics, Tohoku University, Sendai Miyagi 980-8578, Japan
Abstract:
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.
Keywords:
sub-Riemannian geometry, Carnot groups, Popp's volume, systole.
Received: February 10, 2022; in final form July 28, 2022; Published online August 2, 2022
Citation:
Kenshiro Tashiro, “Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume”, SIGMA, 18 (2022), 058, 16 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1854 https://www.mathnet.ru/eng/sigma/v18/p58
|
Statistics & downloads: |
Abstract page: | 44 | Full-text PDF : | 11 | References: | 12 |
|