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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

Citations:

Zbl 0830.53031
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