• Honam Mathematical Journal
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• v.30 no.4
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• pp.693-701
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• 2008

In this paper, when N is a compact Riemannian manifold of class (A), we consider the nonexistence of some warping functions on space-times M = $[a,{\infty}){\times}_fN$ with prescribed scalar curvatures.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.585-594
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• 2022

In this article we classify four dimensional gradient Ricci solitons (M, g, f) with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, (V, g) is isometric to one of the following: (i) an Einstein manifold. (ii) a domain in the Riemannian product (ℝ², g₀) × (N, ${\tilde{g}}$), where g₀ is the flat metric on ℝ² and (N, ${\tilde{g}}$) is a two dimensional Riemannian manifold of constant curvature λ ≠ 0. (iii) a domain in ℝ × W with the warped product metric $ds^2+h(s)^2{\tilde{g}}$, where ${\tilde{g}}$ is a constant curved metric on a three dimensional manifold W.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.41-63
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• 2024

In this paper, we study complete Riemannian immersions into a semi-Riemannian warped product obeying suitable curvature constraints. Under appropriate differential inequalities involving higher order mean curvatures, we establish rigidity and nonexistence results concerning these immersions. Applications to the cases that the ambient space is either an Einstein manifold, a steady state type spacetime or a pseudo-hyperbolic space are given, and a particular investigation of entire graphs constructed over the fiber of the ambient space is also made. Our approach is based on a parabolicity criterion related to a linearized differential operator which is a divergence-type operator and can be regarded as a natural extension of the standard Laplacian.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1255-1274
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• 2008

The purpose of this paper is to discuss the constant term appearing in the BFK-gluing formula for the zeta-determinants of Laplacians on a complete Riemannian manifold when the warped product metric is given on a collar neighborhood of a cutting compact hypersurface. If the dimension of a hypersurface is odd, generally this constant is known to be zero. In this paper we describe this constant by using the heat kernel asymptotics and compute it explicitly when the dimension of a hypersurface is 2 and 4. As a byproduct we obtain some results for the value of relative zeta functions at s=0.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.609-614
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• 1995

Let (M, g) be a complete Riemannian manifold and G be a closed (connected) subgroup of the group of isometries of M. Then the union ${\MM}$ of all principal orbits is an open dense subset of M and the quotient map ${\MM} \longrightarrow {\BB} := {\MM}/G$ becomes a Riemannian submersion for the restriction of g to ${\MM}$ which gives the quotient metric on ${\BB}$. Namely, B is a singular (complete) Riemannian space such that $\partialB$ consists of non-principal orbits.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.367-376
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• 2006

We investigate semi-symmetry and pseudo-symmetry of some 3-dimensional Riemannian manifolds: the D'Atri spaces, the Thurston geometries as well as the ${\eta}$-Einstein manifolds. We prove that all these manifolds are pseudo-symmetric and that many of them are not semi-symmetric.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.757-767
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• 1998

For a Riemannian manifold $(M^n, g)$ we consider the space $V(M^n, g)$ of all smooth functions on $M^n$ whose Hessian is proportional to the metric tensor $g$. It is well-known that if $M^n$ is a space form then $V(M^n)$ is of dimension n+2. In this paper, conversely, we prove that if $V(M^n)$ is of dimension $\ge{n+1}$, then $M^n$ is a Riemannian space form.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.471-484
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• 2005

We study Riemannian or pseudo-Riemannian manifolds which carry the space of closed conformal vector fields of at least 2-dimension. Subject to the condition that at each point the set of closed conformal vector fields spans a non-degenerate subspace of the tangent space at the point, we prove a global and a local classification theorems for such manifolds.