Korean J. Math. Vol. 32 No. 4 (2024) pp.783-790
DOI: https://doi.org/10.11568/kjm.2024.32.4.783

Geodesics on the Kahler cone of the Heisenberg group

Main Article Content

Joonhyung Kim
Ioannis Platis
Li-jie Sun

Abstract

 In this paper, we describe the geodesics on the K\"ahler cone of the Heisenberg group. Furthermore, we also prove that this is not a complete manifold.



Article Details

References

[1] C. P. Boyer, "The Sasakian geometry of the Heisenberg group," Bull. Math. Soc. Sci. Math. Roumanie (N.S.), vol. 52 (100), no. 3, pp. 251–262, 2009. Available: https://www.jstor.org/stable/43679134. Google Scholar

[2] L. Capogna, D. Danielli, S. D. Pauls, and J. T. Tyson, "An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem," Progress in Mathematics, Birkhäuser Verlag, Basel, 2007. Available: https://link.springer.com/book/10.1007/978-3-7643-8133-2. Google Scholar

[3] W. M. Goldman, "Complex hyperbolic geometry," Oxford University Press, 1999. Available: https://global.oup.com/academic/product/complex-hyperbolic-geometry-9780198537939?cc=kr&lang=en. Google Scholar

[4] P. Hajłasz and S. Zimmerman, "Geodesics in the Heisenberg Group," Anal. Geom. Metr. Spaces, vol. 3, no. 1, pp. 325–337, 2015. DOI: https://doi.org/10.1515/agms-2015-0020. Google Scholar

[5] J. Kim, I. D. Platis, and L.-J. Sun, "PCR Kähler equivalent metrics in the Siegel domain," Preprint, arXiv:2304.08079. Available: https://arxiv.org/abs/2304.08079. Google Scholar

[6] V. Marenich, "Geodesics in Heisenberg groups," Geom. Dedicata, vol. 66, no. 2, pp. 175–185, 1997. DOI: https://doi.org/10.1023/A:1004916117293. Google Scholar

[7] G. A. Noskov, "Geodesics in the Heisenberg group: an elementary approach," Sib. Electr. Math. Izv., vol. 5, pp. 177–188, 2008. Available: https://www.mathnet.ru/links/16ac4640d6fec467299dc3a252b69f11/semr97.pdf. Google Scholar

[8] J. R. Parker, "Notes on Complex Hyperbolic Geometry." Available: https://maths.dur.ac.uk/users/j.r.parker/img/NCHG.pdf. Google Scholar

[9] I. D. Platis and L.-J. Sun, "Half plane geometries: zero and unbounded negative curvature," Preprint, arXiv:2111.07569. Available: https://arxiv.org/abs/2111.07569. Google Scholar