Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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VOLUMES OF GEODESIC BALLS IN HEISENBERG GROUPS

  • Jeong, Sunjin (Department of Mathematics University of Ulsan) ;
  • Park, Keun (Department of Mathematics University of Ulsan)
  • Received : 2018.06.04
  • Accepted : 2018.09.01
  • Published : 2018.11.15
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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 $$Vol(B_e(R))={\frac{\pi}{6}}\{-16R+(R^2+6){\sin}\;R+(R^3+10R){\cos}\;R+(R^4+12R^2){\int\nolimits_0^R}\;{\frac{{\sin}\;t}{t}}dt\}$$.

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References

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