• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.1113-1129
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• 2019

We define a class of semi-symmetric metric connection on a Riemannian manifold for which the conformal, the projective, the concircular, the quasi conformal and the m-projective curvature tensors are invariant. We also study the properties of semisymmetric, Ricci semisymmetric and Eisenhart problems for solving second order parallel symmetric and skew-symmetric tensors on the Riemannian manifolds equipped with a semi-symmetric metric P-connection.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1003-1016
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• 2023

In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M_⊥×_ϕM_T, where M_⊥ is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type M_T×_ϕM_⊥ of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.

• The Pure and Applied Mathematics
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• v.7 no.2
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• pp.115-127
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• 2000

The purpose of this paper is to prove the fundamental existence and uniqueness theorems of null curves in semi-Riemannian manifolds M of index 2.

• East Asian mathematical journal
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• v.30 no.3
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• pp.371-383
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• 2014

We study the geometry of lightlike submanifolds of a semi-Riemannian manifold. The purpose of this paper is to prove two singular theorems for irrotational lightlike submanifolds M of a semi-Riemannian space form $\bar{M}(c)$ admitting a semi-symmetric non-metric connection such that the structure vector field of $\bar{M}(c)$ is tangent to M.

• The Pure and Applied Mathematics
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• v.10 no.2
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• pp.119-126
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• 2003

In this paper, when N is a compact Riemannian manifold, we considered the positive time solution to equation $\Box_gu(t,x)-c_nu(t,x)+c_nu(t,x)^{(n+3)/(n-1)}$ on M ＝$(-{\infty},+{\infty})\;{\times}_f\;N$, where $c_n$ ＝(n－1)/4n and $\Box_{g}$ is the d'Alembertian for a Lorentzian warped manifold.

• The Pure and Applied Mathematics
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• v.18 no.4
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• pp.353-359
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• 2011

In this paper, we characterize a semi-Riemannian manifolds satisfies the axiom of indefinite surfaces. We obtain the following result: If a semi-Riemannian manifold satisfies the axiom of indefinite surfaces, then it is a real space form.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.865-879
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• 2022

Let (M^m, g) be an m-dimensional Riemannian manifold. In this paper, we introduce a new class of metric on (M^m, g), obtained by a non-conformal deformation of the metric g. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. In the last section we characterizes some class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when (M^m, g) is an Euclidean space.

• Honam Mathematical Journal
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• v.45 no.2
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• pp.248-268
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• 2023

As a generalization of conformal semi-invariant submersion, conformal slant submersion and conformal semi-slant submersion, in this paper we study conformal hemi-slant submersion from Kenmotsu manifold onto a Riemannian manifold. The necessary and sufficient conditions for the integrability and totally geodesicness of distributions are discussed. Moreover, we have obtained sufficient condition for a conformal hemi-slant submersion to be a homothetic map. The condition for a total manifold of the submersion to be twisted product is studied, followed by other decomposition theorems.

• 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.

• The Pure and Applied Mathematics
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• v.8 no.1
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• pp.33-46
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• 2001

The purpose of this paper is to study totally umbilical coisotropic sub-manifold(M. g, SM) of a semi-Riemannian manifold(M,g)