Differential geometric properties on the Heisenberg group
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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.
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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