• 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.

• The Pure and Applied Mathematics
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• v.5 no.2
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• pp.115-122
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• 1998

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future(or past) complete Lorentzian metrics on $M{\;}={\;}[a,{\;}{\infty}){\times}_f{\;}N$ with specific scalar curvatures.

• The Pure and Applied Mathematics
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• v.6 no.2
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• pp.95-101
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• 1999

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future complete Lorentzian metrics on $M{\;}={\;}[\alpha,\infty){\times}_f{\;}N$ with specific scalar curvatures.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1435-1449
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• 2020

In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.273-284
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• 2001

Let P be a spacelike (n-2)-dimensional submanifold of an n-dimensional Lorentz manifold M and let$\sigma$ be a P-normal null geodesic with Ric($\sigma',\sigma'$)$\geq$m, for the any given nonpositive constant m. We establish a sufficient condition such that there is a focal point of P along $\sigma$.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.857-864
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• 2009

The special conformally flatness is a generalization of a sub-projective space. B. Y. Chen and K. Yano ([4]) showed that every canal hypersurface of a Euclidean space is a special conformally flat space. In this paper, we study the conditions for the base space B is special conformally flat in the conharmonically flat warped product space $B^n{\times}f\;R^1$.