Totally geodesic submanifolds of a trans-Sasakian manifold (Q2871316)

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.