Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

  • Park, Joon-Sik (Department of Mathematics Busan University of Foreign Studies)
  • Received : 2015.08.01
  • Published : 2016.09.01
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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.

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Acknowledgement

Supported by : Busan University of Foreign Studies

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