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.


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
Full Text: DOI arXiv


[1] DOI: 10.4310/MRL.2009.v16.n2.a10 · Zbl 1178.32014
[2] DOI: 10.1007/s00209-009-0643-3 · Zbl 1229.53055
[3] DOI: 10.1007/BF01443359 · Zbl 0666.58042
[4] DOI: 10.1016/j.aim.2003.10.009 · Zbl 1114.53043
[5] DOI: 10.4310/AJM.2002.v6.n2.a5 · Zbl 1127.53304
[6] Friedrich T., J. Reine Angew. Math. 559 pp 217–
[7] Ganchev G., Math. Balkanica (N.S.) 7 pp 261–
[8] DOI: 10.1016/0550-3213(84)90592-3
[9] Gaudochon P., Boll. Unione Mat. Ital. B (7) 11 pp 257–
[10] DOI: 10.1007/s002200000231 · Zbl 0993.53016
[11] DOI: 10.1142/9789812704191_0007
[12] DOI: 10.1007/s00022-007-1947-2 · Zbl 1138.53040
[13] DOI: 10.1016/0370-2693(96)00393-0 · Zbl 1376.53093
[14] DOI: 10.1088/0264-9381/18/6/309 · Zbl 0990.53078
[15] Kobayshi S., Foundations of Differential Geometry (1969) · Zbl 0940.57040
[16] DOI: 10.1016/j.geomphys.2010.09.018 · Zbl 1229.53056
[17] DOI: 10.1142/S0219887811005026 · Zbl 1214.53056
[18] DOI: 10.1142/9789812701763_0016
[19] Mekerov D., C. R. Acad. Bulgare Sci. 61 pp 1249–
[20] Mekerov D., C. R. Acad. Bulgare Sci. 62 pp 1501–
[21] Mekerov D., C. R. Acad. Bulgare Sci. 63 pp 19–
[22] DOI: 10.1016/0550-3213(86)90286-5
[23] DOI: 10.1215/00127094-2010-059 · Zbl 1214.53060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.