DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

Joon-Sik Park
2016 Journal of the Korean Mathematical Society  
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 metrich on the Heisenberg manifold H/Γ such that the Riemannian connection on (H/Γ,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
more » ... trarily 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.
doi:10.4134/jkms.j150453 fatcat:tgk7aw6ekndnje7cprw5p43fmu