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VOLUMES OF GEODESIC BALLS IN HEISENBERG GROUPS ℍ5

  • Kim, Hyeyeon (Department of Mathematics University of Ulsan)
  • Received : 2019.04.30
  • Accepted : 2019.08.13
  • Published : 2019.08.15

Abstract

Let ${\mathbb{H}}^5$ be the 5-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}^5$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}^5$) and radius R in ${\mathbb{H}}^5$. Then, the volume of $B_e(R)$ is given by $${\hfill{12}}Vol(B_e(R))\\{={\frac{4{\pi}^2}{6!}}{\left(p_1(R)+p_4(R){\sin}\;R+p_5(R){\cos}\;R+p_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^R}{\frac{{\sin}\;t}{t}}dt\right.}\\{\left.{\hfill{65}}{+q_4(R){\sin}(2R)+q_5(R){\cos}(2R)+q_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^{2R}}{\frac{{\sin}\;t}{t}}dt}\right)}$$ where $p_n$ and $q_n$ are polynomials with degree n.

References

  1. J. Berndt, F. Tricerri, and L. Vanhecke, Geometry of generalized Heisenberg groups and their Damak-Ricci harmonic extensions, Lecture Notes in Mathematics, 1598 (1995), 51-68.
  2. P. Eberlein, Geometry of 2-step Nilpotent Lie groups with a left invariant metric, Ann. Scient. Ecole Normale Sup., 27 (1994), no. 5 , 611-660.
  3. S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Springer-Verlag, Berlin, 1990.
  4. C. Jang, J. Kim, Y. Kim, and K. Park, Conjugate points on the Quaternionic Heisenberg group, Jour. of Korean Math. Soc., 40 (2003), 61-72.
  5. C. Jang, J. Park, and K. Park, Geodesic Spheres and Balls of the Heisenberg groups, Commun. Korean Math. Soc., 25 (2010), 83-96.
  6. C. Jang and K. Park, Conjugate Points on 2-step Nilpotent Groups, Geom. Dedicata, 79 (2000), 65-80.
  7. S. Jeong and K. Park, Volume of Geodesic Balls in Heisenberg groups, Jour. of the Chungcheong Math. Soc., 31 (2018), no. 4, 369-379. https://doi.org/10.14403/JCMS.2018.31.1.369
  8. A. Kaplan, Riemannian Nilmanifolds attached to Clifford modules, Geom. Dedicata, 11 (1981), 127-136.
  9. A. Kaplan, On the geometry of groups of Heisenberg Type, Bull. London Math. Soc. , 15 (1983), 35-42.
  10. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin, 1972.
  11. G. Walschap, Cut and Conjugate Loci in two-step Nilpotent Lie groups, Jour. of Geometric Analysis, 7 (1997), 343-355.
  12. Wikipedia the Free Encyclopedia, Volume of an n-ball, https://en.wikipedia.org/wiki/Volumeofann-ball.