## Natural connection with totally skew-symmetric torsion on almost contact manifolds with $$B$$-metric.(English)Zbl 1261.53025

An almost contact $$B$$-metric (a.c. $$B$$-m.) manifold is a $$2n+1$$-dimensional manifold $$M$$ endowed with an almost contact structure $$(\varphi,\xi,\eta)$$ and a pseudo-Riemannian metric $$g$$ of signature $$(n,n+ 1)$$ such that $$g(\varphi x,\varphi y)= -g(x, y)+ \eta(x)\eta(y)$$, for any $$x,y\in TM$$.
Given an a.c. $$B$$-m. manifold $$(M,\varphi,\xi,\eta,g)$$, a natural connection on $$M$$ is a linear connection preserving $$g$$ and the almost contact structure. With any natural connection $$D$$ is associated the torsion, namely the $$(0,3)$$-tensor field $$T$$ acting as $$T(x,y,z)= g(D_x y- D_y x-[x, y],z)$$. Then $$D$$ is called a $$\varphi KT$$-connection if $$T$$ is totally skew-symmetric.
Concerning the existence of $$\varphi KT$$-connections, the author proves that the a.c. $$B$$-m. manifold $$(M,\varphi,\xi,\eta,g)$$ admits a $$\varphi KT$$-connection if and only if $$\xi$$ is Killing and $$g((\nabla_x\varphi) y,z)+ g((\nabla_y\varphi)z, x)+g((\nabla_z\varphi) x,y)= 0$$, $$\nabla$$ denoting the Levi-Cività connection. This result allows to relate the existence of $$\varphi KT$$-connections to the classification of a.c. $$B$$-m. manifolds given by G. Ganchev et al. [Math. Balk., New Ser. 7, No. 3–4, 261–276 (1993; Zbl 0830.53031)].
Furthermore, the author studies the curvature tensor of a $$\varphi KT$$-connection. In particular, if $$D$$ is a $$\varphi KT$$-connection with $$D$$-parallel torsion $$T$$, then $$T$$ is a closed 3-form if and only if the curvature of $$D$$ is a $$\varphi$$-Kähler-type tensor. Other properties are obtained in particular cases.
Finally, he constructs a family of 5-dimensional Lie groups admitting a $$\varphi KT$$-connection with $$D$$-parallel torsion.

### MSC:

 53C05 Connections (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Zbl 0830.53031
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