Totally geodesic submanifolds of a trans-Sasakian manifold
type: Item
instance of: scholarly article
title: Totally geodesic submanifolds of a trans-Sasakian manifold
zbMATH Open document ID: 1295.53050
author: Avik De
published in: Proceedings of the Estonian Academy of Sciences
publication date: 2014-01-22T00:00:00Z
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6249202
zbMATH Keywords: invariant submanifold
zbMATH Keywords: trans-Sasakian manifold
zbMATH Keywords: totally geodesic
zbMATH Keywords: semi-parallel
zbMATH Keywords: recurrent
zbMATH Keywords: pseudo-parallel
zbMATH Keywords: Ricci generalized
MaRDI profile type: Publication
full work available at URL: proc.2013.4.05
OpenAlex ID: W2322398815
Wikidata QID: Q124935033
DOI: 10.3176/PROC.2013.4.05
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review text: Invariant submanifolds of a contact manifold have been a major area of research for a long time since the concept was borrowed from complex geometry, which helps us to understand several important topics of applied mathematics; for example, in studying non-linear autonomous systems the idea of invariant submanifolds plays an important role. Recall that a submanifold of a contact manifold is said to be totally geodesic if every geodesic in that submanifold is also geodesic in the ambient manifold. Note that several studies have been done on invariant submanifolds of trans-Sasakian manifolds, which can be regarded as a generalization of Sasakian, Kenmotsu, and cosymplectic structures. Specially, \textit{A. Sarkar} and \textit{M. Sen} [Proc. Est. Acad. Sci. 61, No. 1, 29--37 (2012; Zbl 1244.53024)] recently proved some equivalent conditions of an invariant submanifold of trans-Sasakian manifolds to be totally geodesic.NEWLINENEWLINENEWLINEIn the present paper, the authors rectify proofs of most of the major theorems of [loc. cit.] and \textit{A. Turgut Vanlı} and \textit{R. Sari} [Differ. Geom. Dyn. Syst. 12, 277--288 (2010; Zbl 1200.53019)], show some theorems of [Sarkar et al., loc. cit.] as corollary of their present results. Moreover, they also study invariant submanifolds of a trans-Sasakian manifold satisfying \(Z\left( {X,Y} \right).h = 0\), where \(Z\) is the concircular curvature tensor. Finally, the authors list some new and useful equivalent conditions for an invariant submanifold of a trans-Sasakian manifold to be totally geodesic.
reviewed by: Xiang Gao
description: scientific article; zbMATH DE number 6249202