On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric. (English) Zbl 1224.53059

Mekerov, Dimitar

C. R. Acad. Bulg. Sci. 63, No. 1, 19-28 (2010).

Let \((M, J, g)\) be a \(2n\)-dimensional almost complex manifold with Norden metric, i.e. for all tangent vector fields \(X\), \(Y\) on \(M\) the almost complex structure \(J\) and the metric \(g\) of signature \((n,n)\) satisfy the conditions \(J^2 X=-X\), \(g(J X, J Y)=-g(X,Y)\). A linear connection \(\nabla'\) on \((M, J, g)\) is natural if it preserves the almost complex structure \(J\) and the metric \(g\), i.e. \(\nabla' J=\nabla' g=0\). The author proves that if an almost complex manifold with Norden metric admits a natural connection with skew-symmetric torsion tensor, then it is a quasi-Kähler manifold with Norden metric.

Reviewer: Veselka Mihova (Sofia)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B05	Linear and affine connections
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

almost complex manifold with Norden metric; natural linear connection; totally skew-symmetric torsion