• 대한수학회논문집
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• 제9권4호
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• pp.917-926
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• 1994

Let ($M, G_M, F$) be a (p+q)-dimensional Riemannian manifold with a foliation F of codimension q and a bundle-like metric $g_M$ with respect to F ([9]). Aside from the Laplacian $\bigtriangleup_g$ associated to the metric g, there is another differnetial operator, the Jacobi operator $J_D$, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum isdiscrete as a consequence of the compactness of M. The study of the spectrum of $\bigtriangleup_g$ acting on functions or forms has attracted a lot of attention. In this point of view, the present authors [7] have studied the spectrum of the Laplacian and the curvature of a compact orientable cosymplectic manifold. On the other hand, S. Nishikawa, Ph. Tondeur and L. Vanhecke [6] studied the spectral geometry for Riemannian foliations. The purpose of the present paper is to study the relation between two spectra and the transversal geometry of cosymplectic foliations. We shall be in $C^\infty$-category. Manifolds are assumed to be connected.

• 대한수학회논문집
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• 제35권2호
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• pp.613-623
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• 2020

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.

• 한국수학사학회지
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• 제28권2호
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• pp.103-115
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• 2015

Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and Levi-Civita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.

• 대한수학회보
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• 제56권5호
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• pp.1219-1233
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• 2019

Let $E{\rightarrow}M$ be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection $D^E$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when $D^E$ is flat. We study also the Einstein property on E proving, among other results, that if $k{\geq}2$ and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.

• 대한수학회보
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• 제40권3호
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• pp.475-487
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• 2003

A generalized even-dimensional Riemannian manifold defined by the ME-connection which is both Einstein and of the form (3.3) is called an even-dimensional ME-manifold and we denote it by $MEX_{2n}$. The purpose of this paper is to study a necessary and sufficient condition that there is an ME-connection, to derive the useful properties of some tensors, and to investigate a representation of the ME-vector in $MEX_{2n}$.

• Kyungpook Mathematical Journal
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• 제46권3호
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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.

• 대한수학회보
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• 제49권6호
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• pp.1163-1178
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• 2012

In this paper, we study the geometry of Einstein half lightlike submanifolds M of a semi-Riemannian space form $\bar{M}(c)$ subject to the conditions: (a) M is screen conformal, and (b) the coscreen distribution of M is a conformal Killing one. The main result is a classification theorem for screen conformal Einstein half lightlike submanifolds of a Lorentzian space form with a conformal Killing coscreen distribution.

• 대한수학회보
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• 제54권2호
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• pp.619-632
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• 2017

We define a new connection on semi-Riemannian manifold, which is called a non-metric ${\phi}$-symmetric connection. Semi-symmetric non-metric connection and quarter-symmetric non-metric connection are two impotent examples of this connection. The purpose of this paper is to study the geometry of lightlike hypersurfaces of an indefinite Kaehler manifold with a non-metric ${\phi}$-symmetric connection.

• 대한수학회논문집
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• 제25권2호
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• pp.225-234
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• 2010

In this paper, we study the geometry of lightlike hypersurfaces of a semi-Riemannian manifold. We prove a classification theorem for Einstein lightlike hypersurfaces M of a Lorentzian space form subject such that the second fundamental forms of M and its screen distribution S(TM) are conformally related by some non-vanishing smooth function.

• 대한수학회논문집
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• 제31권3호
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• pp.613-624
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• 2016

We define a new connection on a semi-Riemannian manifold. Its notion contains two well known notions; (1) semi-symmetric connection and (2) quarter-symmetric connection. In this paper, we study the geometry of lightlike hypersurfaces of an indefinite generalized Sasakian space form with a symmetric metric connection of type (${\ell}$, m).