We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem by Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.
In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with a metric structure. Given a pseudo-Riemannian metric structure, we describe symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds. Especially, we discuss the first canonical Hermitian connection and show the formula for it in the case of nearly Kaehler manifolds using the p…
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…
We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space considered with a natural family of Riemannian metrics defined by means of the metric and the given connection on the base manifold.
${\mathrm G}_2$-structures with parallel skew-symmetric torsion
Université Paris-Saclay, CNRS（巴黎萨克雷大学，法国国家科学研究中心）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛数学研究所）；Universität Stuttgart（斯图加特大学）
The paper is devoted to metric connections with parallel skew-symmetric torsion in Lorentzian signature. This is motivated by recent progress in the Riemannian signature and by possible applications to supergravity theories. We provide a complete information about holonomy algebras, torsion and curvature of the considered connections up to the corresponding objects from the Riemannian signature. Various examples are constructed. It is shown how to construct all simply connected Lorentzian naturally reductive homogeneous spaces of arbitrary dimension from Riemannian naturally reductive homogene…
Geometries with parallel, skew-symmetric and closed torsion
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；Universidade Estadual de Campinas（坎皮纳斯州立大学）
On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric
本文考虑了具有Norden度量的殆复流形（即. e.相对于该度量，殆复结构是反等距的）。在这样的流形上，我们研究了保持殆复结构和度量的线性联络，它具有全斜对称的挠率张量（即：e. a 3-形式）。证明了若一个具有Norden度量的非Kaehler殆复流形容许这种联络，则该流形是拟Kaehler流形（即，若该流形是拟Kaehler流形，则该流形是拟Kaehler流形）。e.具有不可积的殆复结构）。我们证明了这种联系是唯一的，确定了它的形式，并构造了一个李群上的例子。我们考虑的情况下，当流形允许一个连接与平行全斜对称挠和的情况下，当这种连接有Kaehler曲率张量。得到了迷向Kaehler流形具有Norden度量的充要条件。
Adapted connections with skew-torsion on metric $f$-manifolds
Institute of Mathematics, Jagiellonian University in Kraków（克拉科夫雅盖隆大学数学研究所）；Department of Mathematics and Statistics, Faculty of Science, Masaryk University（马萨里克大学理学院数学与统计系）
This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar curvature, it is necessarily compact and it has finite first fundamental group. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with s…
University of Sofia, Faculty of Mathematics and Informatics（索非亚大学数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息研究所）
