×

On some special type of trans-Sasakian manifolds. (English) Zbl 1205.53029

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.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B20 Local Riemannian geometry
53B05 Linear and affine connections
PDFBibTeX XMLCite