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Abstract. We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the Vparallel spinors. In particular, we obtain partial solutions of the type // string equations in dimension n = 5, 6 and 7.

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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.

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Abstract. We compute the scalar curvature of 7-dimensional G 2 -manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n = 7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2 -structure into a cocalibrated one of pure type W 3 .

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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.