Abstract. We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N (1, 1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits P 2 -parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.

Given a reductive homogeneous space M = G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ t joining the canonical and the Levi-Civita connection (t = 0, 1/2). We show that the Dirac operator D t corresponding to t = 1/3 is the so-called "cubic" Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator D on spinors and an eigenvalue estimate for the first eigenvalue of D 1/3 . This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T = 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and give a detailed discussion of some 5-dimensional example.

a b s t r a c tIt is well known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G 2 -structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G 2 -structure in this family whose characteristic connection ∇ c along with its parallel spinor field Ψ 0 can be used for a thorough investigation of the geometric properties of 7-dimensional 3-Sasakian manifolds. Many known and some new properties can be easily derived from the properties of ∇ c and of Ψ 0 , yielding thus an appropriate substitute for the missing characteristic connection.

We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ∇ with skew torsion T ∈ Λ 3 M by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich's classical Riemannian estimate. We also determine a novel twistor and Killing equation with torsion and use it to discuss the case in which the minimum is attained in the bound.

We present a new method for classifying naturally reductive homogeneous spaces -i. e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions 3, 4, and 5, and yields as a new result the classification in dimension 6. In each dimension, one obtains a 'hierarchy' of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a 'generic' parallel torsion form defines a quasi-Sasakian structure in dimension 5 and an almost complex structure in dimension 6. within the priority programme 1388 "Representation theory". Ana Ferreira thanks Philipps-Universität Marburg for its hospitality during a research stay in May-July 2013, and she also acknowledges partial financial support by the FCT through the project PTDC/MAT/118682/2010 and the University of Minho through the FCT project PEst-C/MAT/UI0013/2011. We also thank Andrew Swann (Aarhus) for discussions on Section 4 during a research visit to Marburg in June 2013 and Simon G. Chiossi (Marburg) for many valuable comments on a preliminary version of this work. Metric connections with skew torsionConsider a Riemannian manifold (M n , g). The difference between its Levi-Civita connection ∇ g and any linear connection ∇ is a (2, 1)-tensor field A,The curvature of ∇ resp. ∇ g will always be denoted by R resp. R g . Following Cartan, we study the algebraic types of the torsion tensor for a metric connection. Denote by the same symbol the (3, 0)-tensor derived from a (2, 1)-tensor by contraction with the metric. We identify T M n with (T M n ) * using g from now on. Let T be the n 2 (n − 1)/2-dimensional space of all possible torsion tensors,A connection ∇ is metric if and only if A belongs to the spaceThe spaces T and A are isomorphic as O(n) representations, reflecting the fact that metric connections can be uniquely characterized by their torsion. For n ≥ 3, they split under the action of O(n) into the sum of three irreducible representations,