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.

Abstract. Consider the nonstandard embedding of SO(3) into SO(5) given by the 5-dimensional irreducible representation of SO(3), henceforth called SO(3) ir . In this note, we study the topology and the differential geometry of 5-dimensional Riemannian manifolds carrying such an SO(3) ir structure, i. e. with a reduction of the frame bundle to SO(3) ir .

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