Prasad, Rajendra; Pankaj; Tripathi, M. M.; Shukla, S. S. On some special type of trans-Sasakian manifolds. (English) Zbl 1205.53029 Riv. Mat. Univ. Parma (8) 2, 1-17 (2009). Let \(M\) be a \((2n+1)\)-dimensional almost contact metric manifold equipped with almost contact metric structure \((\phi , \xi , \eta , g)\), where \(\phi \) is (1,1) tensor field, \(\xi \) is a vector field, \(\eta \) is 1-form and \(g\) is compatible Riemann metric such that \[ \phi ^2=-I +\eta \otimes \xi,\quad \eta (\xi )=1,\quad \phi \xi =0,\quad \eta \circ \phi =0, \]\[ g(\phi X, \phi Y )=g(X,Y)-\eta (X)\eta (Y), \]\[ g(\phi X,Y)=-g(X,\phi Y),\quad g(X,\xi )=\eta (X), \]\(\forall X,Y \in TM.\) A manifold \(M\) is called a trans-Sasakian manifold if\[ (\nabla _X\phi )Y=\alpha \{g(X,Y)\xi -\eta (Y)X\}+\beta \{g(\phi X,Y)\xi -\eta (Y)\phi X\}, \]where \(\nabla \) is the Levi-Civita connection of the Riemannian metric \(g\) and \(\alpha \) and \(\beta \) are smooth functions on \(M\). The present paper is focused on the study of 3-dimensional trans-Sasakian manifolds. Necessary and sufficient conditions for a trans-Sasakian manifold of type \((\alpha , \beta )\) to be \(\eta\)-Einstein are given. It is also shown that every 3-dimensional \(\alpha\)-Sasakian, \(\beta\)-Kenmotsu and \((\alpha , \beta )\) trans-Sasakian manifold, where \(\alpha \) and \(\beta \) are constants, are always \(\eta\)-Einstein.It is proved that 3-dimensional conharmonically flat trans-Sasakian manifolds have zero constant curvatures.Also, projectively flat trans-Sasakian manifolds are studied. Reviewer: Iulia Hirică (Bucureşti) Cited in 3 Documents MSC: 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53B20 Local Riemannian geometry 53B05 Linear and affine connections Keywords:trans-Sasakian manifold; conharmonically flat; conformally flat; projectively flat; \(\eta \)-Einstein manifold; curvature tensor; Ricci curvature; Ricci operator PDFBibTeX XMLCite \textit{R. Prasad} et al., Riv. Mat. Univ. Parma (8) 2, 1--17 (2009; Zbl 1205.53029)