• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.553-562
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• 1995

In a Recent paper [3], D. Chinea, M. Delon and J. C. Marrero proved that a cosymplectic manifold is formal and constructed an example of compact cosymplectic manifold which is not a global product of a Kaehler manifold with the circle. In this paper we study the compact cosymplectic manifolds with critical Riemannian metrics.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.421-431
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• 1998

We characterize real 4-dimensional Kahler metrices which are critical for natural quadratic Riemannian functionals defined on the space of all Riemannian metrics. In particular we show that such critical Kahler surfaces are either Einstein or have zero scalar curvature. We also make some discussion on criticality in the space of Kahler metrics.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.109-127
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• 2007

We classify complete, locally homogeneous metrics with finite volume on four-dimensional manifolds which are critical points for the squared $L^2-norm$ functionals of either the full Riemannian curvature tensor or the Weyl curvature tensor defined on the space of Riemannian metrics.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.347-354
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• 1997

The conharmonic transforamtion is a conformal transformation which satisfies a specified differential equation. Such a transformation was defined by Y. Ishi and we generalize his results. In particular, we obtain a necessary and sufficient condition for the invariance of critical Riemannian metrics under the conharmonic transformation.

• 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.

• Journal of the Korean Mathematical Society
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• v.31 no.2
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• pp.205-211
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• 1994

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1367-1382
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• 2020

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moreover, we also provide a few rigidity results for locally conformally flat critical metrics.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.551-564
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• 2001

We view Weyl structures as generalizations of Riemannian metrics and study the critical points of geometric functional which involve scalar curvature, defined on the space of Weyl structures on a closed 4-manifold. The main goal here is to provide a framework to analyze critical Weyl structures by defining functionals, discussing function spaces and writing down basic formulas for the equations of critical points.

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.13-18
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• 1994

The study of the integral of the scalar curvature, $A(g)\;=\;{\int}_M\;RdV_9$ as a functional on the set M of all Riemannian metrics of the same total volume on a compact orient able manifold M is now classical, dating back to Hilbert [6] (see also Nagano [8]). Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric.(omitted)

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.655-667
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• 2012

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$. We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an $n$-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.