• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.143-165
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• 2023

For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.195-203
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• 2023

Special examples of harmonic manifolds that are not symmetric, proving that the conjecture posed by Lichnerowicz fails in the non-compact case have been intensively studied. We completely classify homogeneous structures on Damek-Ricci spaces equipped with the left invariant metric.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.651-684
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• 2022

Let G be an arbitrary, connected, simply connected and unimodular Lie group of dimension 3. On the space 𝔐(G) of left-invariant Lorentzian metrics on G, there exists a natural action of the group Aut(G) of automorphisms of G, so it yields an equivalence relation ≃ on 𝔐(G), in the following way: h₁ ≃ h₂ ⇔ h₂ = 𝜙^*(h₁) for some 𝜙 ∈ Aut(G). In this paper a procedure to compute the orbit space Aut(G)/𝔐(G) (so called moduli space of 𝔐(G)) is given.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1055-1059
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• 1996

We characterize the left invariant Riemannian metrics on SO(1,2) which give rise to 3- or 4-dimensional isometry groups.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.193-203
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• 2002

Weyl structure can be viewed as generalizations of Riemannian metrics. We study Weyl structures which are critical points of the squared L$^2$ norm functional of the full curvature tensor, defined on the space of Weyl structures on a compact 4-manifold. We find some relationship between these critical Weyl structures and the critical Riemannian metrics. Then in a search for homogeneous critical structures we study left-invariant metrics on some solv-manifolds and prove that they are not critical.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.255-261
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• 2009

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1047-1067
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• 1996

In this paper, we are interested in left invariant flat affine structures on Lie groups. These structures has been studied by many authors in different contexts. One of the fundamental questions is the existence of complete affine structures for solvable Lie groups G, raised by Minor [15]. But recently Benoist answered negatively even for the nilpotent case [1]. Also moduli space of such structures for lower dimensional cases has been studied by several authors, sometimes with compatible metrics [5,10,4,12].

• Honam Mathematical Journal
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• v.38 no.2
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• pp.359-366
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• 2016

Let H be the 3-dimensional Heisenberg group, ($G=H{\times}S^1$, g) a product Riemannian manifold of Riemannian manifolds H and S with arbitrarily given left invariant Riemannian metrics respectively, and ${\Gamma}$ the discrete subgroup of G with integer entries. Then, on the Riemannian manifold ($M:=G/{\Gamma}$, ${\Pi}^*g=\bar{g}$), ${\Pi}:G{\rightarrow}G/{\Gamma}$, we evaluate the scalar curvature and the Ricci curvature.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.115-141
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• 2023

We study a class of left-invariant pseudo-Riemannian Sasaki metrics on solvable Lie groups, which can be characterized by the property that the zero level set of the moment map relative to the action of some one-parameter subgroup {exp tX} is a normal nilpotent subgroup commuting with {exp tX}, and X is not lightlike. We characterize this geometry in terms of the Sasaki reduction and its pseudo-Kähler quotient under the action generated by the Reeb vector field. We classify pseudo-Riemannian Sasaki solvmanifolds of this type in dimension 5 and those of dimension 7 whose Kähler reduction in the above sense is abelian.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1121-1137
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• 2007

There are 7 types of 4-dimensional solvable Lie groups of the form ${\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}$ which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups $({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)$ are well known to have lattices. All the compact forms modeled on the remaining four solvable groups $Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^'4\;and\;Sol_{\lambda}^4$ are characterized: (1) $Sol^3\;{\times}\;{\mathbb{R}}$ has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, ${\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4$. (2) Only some of $Sol_{\lambda}^4$, called $Sol_{m,n}^4$, have lattices with no non-trivial infra-solvmanifolds. (3) $Sol_0^{'4}$ does not have a lattice nor a compact form. (4) $Sol_0^4$ does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on $Sol_0^4$. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.