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http://dx.doi.org/10.14403/jcms.2018.31.1.369

VOLUMES OF GEODESIC BALLS IN HEISENBERG GROUPS  

Jeong, Sunjin (Department of Mathematics University of Ulsan)
Park, Keun (Department of Mathematics University of Ulsan)
Publication Information
Journal of the Chungcheong Mathematical Society / v.31, no.4, 2018 , pp. 369-379 More about this Journal
Abstract
Let ${\mathbb{H}}_3$ be the 3-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}_3$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}_3$) and radius R in ${\mathbb{H}}_3$. Then, the volume of $B_e(R)$ is given by $$Vol(B_e(R))={\frac{\pi}{6}}\{-16R+(R^2+6){\sin}\;R+(R^3+10R){\cos}\;R+(R^4+12R^2){\int\nolimits_0^R}\;{\frac{{\sin}\;t}{t}}dt\}$$.
Keywords
Heisenberg group; geodesic ball;
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