• Korean Journal of Mathematics
• /
• 제23권2호
• /
• pp.313-321
• /
• 2015

The manifold $^*g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{{\lambda}{\nu}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 3-dimensional $^*g-ESX_3$ and to display a surveyable tnesorial representation of 3-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

• Journal of the Korean Mathematical Society
• /
• 제40권1호
• /
• pp.129-167
• /
• 2003

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concircular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in [18] and [19].

• Bulletin of the Korean Mathematical Society
• /
• 제38권2호
• /
• pp.223-230
• /
• 2001

In this parper, we considered the uniqueness of positive time-solution to equation ${\Box}_g$u(t,$\chi$) - $c_n$u(t,$\chi$) + $c_n$u(t,$\chi$)$^[\frac{n+3}{n-3}]$ = 0, where $c_n$ = $\frac{n-1}{4n}$ and ${\Box}_g$ is the d'Alembertian for a Lorentzian warped manifold M = {a,$\infty$] $\times_f$ N.

• Korean Journal of Mathematics
• /
• 제18권2호
• /
• pp.133-139
• /
• 2010

The manifold $g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $g_{{\lambda}{\mu}}$ through the ES-connection which is both Einstein and semi-symmetric. In this paper, we investigate the properties of the vectors $X_{\lambda}$, $S_{\lambda}$ and $U_{\lambda}$ of $g-ESX_n$, with main emphasis on the derivation of several useful generalized identities involving it.

• Kyungpook Mathematical Journal
• /
• 제46권3호
• /
• pp.367-376
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• 제32권4호
• /
• pp.1047-1065
• /
• 2017

The notion of a non-metric ${\phi}$-symmetric connection on semi-Riemannian manifolds was introduced by Jin [6, 7]. The object of study in this paper is generic lightlike submanifolds of an indefinite Kaehler manifold ${\bar{M}}$ with a non-metric ${\phi}$-symmetric connection. First, we provide several new results for such generic lightlike submanifolds. Next, we investigate generic lightlike submanifolds of an indefinite complex space form ${\bar{M}}(c)$ with a non-metric ${\phi}$-symmetric connection.

• Kyungpook Mathematical Journal
• /
• 제48권2호
• /
• pp.183-194
• /
• 2008

We prove that totally umbilical submanifold M of an extended quasi-recurren manifold is also extended quasi-recurrent. If, moreover, M is conformally flat then, locally, M is isometric to the manifold with known metric. Some curvature properties of such submanifold are investigated. Making use of these results we shall prove the existence of totally umbilical submanifold being pseudosymmetric in the sense of Ryszard Deszcz and satisfying some other curvature conditions.

• Communications of the Korean Mathematical Society
• /
• 제31권3호
• /
• pp.613-624
• /
• 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).

• Korean Journal of Mathematics
• /
• 제26권1호
• /
• pp.43-51
• /
• 2018

The manifold $^{\ast}g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^{\ast}g^{{\lambda}{\nu}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to derive a new set of powerful recurrence relations and to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 5-dimensional $^{\ast}g-ESX_5$ and to display a surveyable tnesorial representation of 5-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the second class.

• Communications of the Korean Mathematical Society
• /
• 제35권4호
• /
• pp.1203-1219
• /
• 2020

Jin [7] defined a new connection on semi-Riemannian manifolds, which is a non-symmetric and non-metric connection. He said that this connection is an (ℓ, m)-type connection. Jin also studied lightlike hypersurfaces of an indefinite trans-Sasakian manifold with an (ℓ, m)-type connection in [7]. We study further the geometry of this subject. In this paper, we study generic lightlike submanifolds of an indefinite trans-Sasakian manifold endowed with an (ℓ, m)-type connection.