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Curvature properties of some three-dimensional almost contact manifolds with $B$-metric. II. (English) Zbl 1086.53046
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5–12, 2003. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-8-7/pbk). 169-177 (2004).
A $B$-metric on a $\left(2n+1\right)$-dimensional almost contact manifold $\left(M,\xi ,\eta ,\phi \right)$ is a metric $g$ such that $g\left(\phi X,\phi Y\right)=-g\left(X,Y\right)+\eta \left(X\right)\eta \left(Y\right)$. $g$ is an indefinite metric of signature $\left(n,n+1\right)$. A decomposition of the class of almost contact manifolds with $B$-metric was given by G. Gachev, V. Mihova and K. Gribachev [Math. Balk., New Ser. 7, No. 3–4, 264–276 (1993; Zbl 0830.53031)], where eleven basic classes are defined. In this paper the authors study curvature properties and geometric characteristics of a 3-dimensional almost contact manifold with a $B$-metric belonging to two main classes.
##### MSC:
 53C15 Differential geometric structures on manifolds 53D15 Almost contact and almost symplectic manifolds