scholarly journals On Submersion of CR-Submanifolds of l.c.q.K. Manifold

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Majid Ali Choudhary ◽  
Mahmood Jaafari Matehkolaee ◽  
Mohd. Jamali
Keyword(s):  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Kaehler Manifold ◽  
Almost Hermitian Manifold ◽  
Cr Submanifold ◽  
Locally Conformal ◽  
Cr Submanifolds

We study submersion of CR-submanifolds of an l.c.q.K. manifold. We have shown that if an almost Hermitian manifold B admits a Riemannian submersion π:M→B of a CR-submanifold M of a locally conformal quaternion Kaehler manifold M¯, then B is a locally conformal quaternion Kaehler manifold.

scholarly journals TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A NEARLY KAEHLER MANIFOLD

1996 ◽  
Vol 27 (2) ◽  
pp. 145-149
Author(s):  
S. H. KON ◽  
SIN-LENG TAN
Keyword(s):  
Kaehler Manifold ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Cr Submanifold ◽  
Nearly Kaehler Manifold ◽  
Totally Real ◽  
Cr Submanifolds

The geometry of a CR-submanifold in a Kaehler manifold has been extensively studied. B.Y . Chen has classified the totally umbilical CR-submanifolds of a Kaehler manifold and showed that they are either totally geodesic, or totally real or dim$(D^{\perp}) =1$. In this paper we show that such a result is also true in a nearly Kaehler manifold.

scholarly journals CR-SUBMANIFOLDS OF A QUASl-KAEHLER MANIFOLD

1995 ◽  
Vol 26 (3) ◽  
pp. 261-266
Author(s):  
S. H. KON ◽  
SIN-LENG TAN
Keyword(s):  
Sufficient Conditions ◽  
Kaehler Manifold ◽  
Complex Submanifold ◽  
Cr Submanifold ◽  
Almost Complex ◽  
Cr Submanifolds

Let $M$ be a CR-submanifold of a quasi-Kaehler manifold $N$. Sufficient conditions for the holomorphic distribution $D$ in $M$ to be integrable are derived. We also show that $D$ is minimal. It follows that an (almost) complex submanifold of a quasi-Kaehler manifold is minimal, this generalizes the well known result that a complex submanifold of a Kaehler manifold is minimal.

scholarly journals Twisted product CR-submanifolds in a locally conformal Kaehler manifold

2013 ◽  
Vol 94 (108) ◽  
pp. 131-140
Author(s):  
Koji Matsumoto ◽  
Zerrin Şentürk
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Twisted Product ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Locally Conformal ◽  
Cr Submanifolds

Recently, we have researched certain twisted product CR-submanifolds in a Kaehler manifold and some inequalities of the second fundamental form of these submanifolds [11]. We consider here two kinds of twisted product CR-submanifolds (the first and the second kind) in a locally conformal Kaehler manifold. In these submanifolds, we give inequalities of the second fundamental form (see Theorems 5.1 and 5.2) and consider the equality case of these.

scholarly journals Contact-Complex Riemannian Submersions

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2996
Author(s):  
Cornelia-Livia Bejan ◽  
Şemsi Eken Meriç ◽  
Erol Kılıç
Keyword(s):  
Riemannian Submersion ◽  
Ricci Soliton ◽  
Hermitian Manifold ◽  
Horizontal Distribution ◽  
The Other ◽  
Riemannian Submersions ◽  
Almost Hermitian Manifold ◽  
Base Manifold ◽  
Contact Complex ◽  
Contact Riemannian Manifold

A submersion from an almost contact Riemannian manifold to an almost Hermitian manifold, acting on the horizontal distribution by preserving both the metric and the structure, is, roughly speaking a contact-complex Riemannian submersion. This paper deals mainly with a contact-complex Riemannian submersion from an η-Ricci soliton; it studies when the base manifold is Einstein on one side and when the fibres are η-Einstein submanifolds on the other side. Some results concerning the potential are also obtained here.

scholarly journals TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A KAEHLER MANIFOLD

1993 ◽  
Vol 24 (1) ◽  
pp. 43-49
Author(s):  
S. M. KHURSHEED HAIDER ◽  
V. A. KHAN ◽  
S . I. HUSAIN
Keyword(s):  
Classification Theorem ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
Cr Submanifold ◽  
Cr Submanifolds

In the present paper we study totally umbilical CR-submanifolds of a Kaehler manifold. A classification theorem for a $D^\perp$-totally umbilical CR-submanifold is proved. The conditions under which a CR- submanifold becomes a CR-product are obtained, and finally a theorem for a CR-submanifold to be a proper CR-product is also established.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Space Form ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Locally Conformal

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}. In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}. Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form. In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold. Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}. In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form. Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}. Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$ satisfy some special equations, see Theorem~\ref{th5.2}.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Nearly Kaehler Manifold

In 1994, in [13], N. Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of CR- and slant-submanifolds. In particular, he considered this submanifold in Kaehlerian manifolds, [13]. Then, in 2007, V. A. Khan and M. A. Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, [11]. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and gave a necessary and sufficient conditions for two distributions (holomorphic and slant) to be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form, [4]. In this paper, we define 2-kind warped product semi-slant submanifolds in a locally conformal Kaehler manifold and consider some properties of these submanifolds.

scholarly journals On locally conformal Kaehler space forms

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 593-597
Author(s):  
Pegah Mutlu ◽  
Zerrin Sentürk
Keyword(s):  
Curvature Tensor ◽  
Space Form ◽  
Hermitian Manifold ◽  
Kaehler Manifold ◽  
Space Forms ◽  
Holomorphic Curvature ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Locally Conformal

The notion of a locally conformal Kaehler manifold (an l.c.K-manifold) in a Hermitian manifold has been introduced by I. Vaisman in 1976. In [2], K. Matsumoto introduced some results with the tensor Pij is hybrid. In this work, we give a generalisation about the results of an l.c.K-space form with the tensor Pij is not hybrid. Moreover, the Sato?s form of the holomorphic curvature tensor in almost Hermitian manifolds and l.c.K-manifolds are presented.

scholarly journals On Quasi-Hemi-Slant Riemannian Submersion

2019 ◽  
pp. 1-14
Author(s):  
S. Longwap ◽  
F. Massamba ◽  
N. E. Homti
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Riemannian Submersions ◽  
Almost Hermitian Manifold ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds

We recall the notions of invariant, anti-invarian, semi-invariant, slant, semi-slant, quasi-slant and hemi-slant Riemannian submersions from almost Hermitian manifolds to a Riemannian manifolds. In this paper we contruct a Riemannian submersion which generalizes hemi-slant, semi-slant and semi-invariant Riemanian submersions from almost Hermitian manifold to a Riemannian manifold and study its geometry.

scholarly journals TANGENT BUNDLE ENDOWED WITH QUARTER-SYMMETRIC NON-METRIC CONNECTION ON AN ALMOST HERMITIAN MANIFOLD

2020 ◽  
Vol 35 (1) ◽  
pp. 167
Author(s):  
Mohammad Nazrul Islam Khan
Keyword(s):  
Tangent Bundle ◽  
Hermitian Manifold ◽  
Nijenhuis Tensor ◽  
Kaehler Manifold ◽  
Almost Hermitian Manifold ◽  
Base Manifold ◽  
Metric Connection

In this paper, we have studied the tangent bundle endowed with quarter-symmetric non-metric connection obtained by vertical and complete lifts of a quarter-symmetric non-metric connection on the base manifold and, also, proposed the study of the tangent bundle of an almost Hermitian manifold and an almost Kaehler manifold. Finally, we obtained some theorems for Nijenhuis tensor on the tangent bundle of an almost Hermitian manifold and an almost Kaehler manifold.\\

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