[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.j150453

DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

Park, Joon-Sik (Department of Mathematics Busan University of Foreign Studies)

Publication Information

Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1149-1165 More about this Journal

Abstract

In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ( $H/{\Gamma},\bar{h}$ ) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

Keywords

Heisenberg group; Heisenberg manifold; (locally) symmetric Riemannian manifold; Yang-Mills connection; harmonic map; affine map;

Citations & Related Records

Reference

1	J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS, Regional Conf. Ser. in Math. 50, Amer. Math. Soc., Providence, Rhode Island, 1983.
2	S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
3	U.-H. Ki and J.-S. Park, Affine inner automorphisms of SU(2), Tsukuba J. Math. 23 (1999), no. 3, 485-493. DOI
4	S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Wiley-Interscience, New York, 1963.
5	Y. Matsushima, Differentiable Manifolds, Marcel Dekker, Inc. New York, 1972.
6	J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), no. 3, 293-329. DOI
7	J.-S. Park, Critical homogeneous metrics on the Heisenberg manifold, Interdiscip. Inform. Sci. 11 (2005), no. 1, 31-34. DOI
8	J.-S. Park, Yang-Mills connections with Weyl structure, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 129-132. DOI
9	J.-S. Park, Projectively flat Yang-Mills connections, Kyushu J. Math. 64 (2010), no. 1, 49-58. DOI
10	J.-S. Park, Invariant Yang-Mills connections with Weyl structure, J. Geometry Physics 60 (2010), no. 12, 1950-1957. DOI
11	J.-S. Park, Curvatures on SU(3)/T(k, l), Kyushu J. Math. 67 (2013), no. 1, 55-65. DOI
12	J.-S. Park and W.-T. Oh, The Abbena-Thurston manifold as a critical point, Canad. Math. Bull. 39 (1996), no. 3, 352-359. DOI
13	H. Urakawa, Calculus of Variations and Harmonic Maps, Amer. Math. Soc., Providence, Rhode Island, 1993.
14	M. Wang and W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986), no. 1, 177-194. DOI
15	J. A. Wolf, Curvatures in nilpotent Lie groups, Proc. Amer. Math. Soc. 15 (1964), 271-274. DOI