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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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참고문헌 (15)
참고문헌 신청
J. Eells / 1983 / Selected topics in harmonic maps / Amer. Math. Soc
S. Helgason / 1978 / Differential Geometry, Lie Groups and Symmetric Spaces / Academic Press
U.-H. Ki / 1999 / Affine inner automorphisms of SU(2) / Tsukuba J. Math 23(3):485~493
S. Kobayashi / 1963 / Foundations of Differential Geometry. Vol. I / WileyInterscience
Y. Matsushima / 1972 / Differentiable Manifolds / Marcel Dekker, Inc
S. Helgason / 1978 / Differential Geometry, Lie Groups and Symmetric Spaces / Academic Press
U.-H. Ki / 1999 / Affine inner automorphisms of SU(2) / Tsukuba J. Math 23(3):485~493
S. Kobayashi / 1963 / Foundations of Differential Geometry. Vol. I / WileyInterscience
Y. Matsushima / 1972 / Differentiable Manifolds / Marcel Dekker, Inc
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