The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.

In 1986, R. D. Cook proposed differential geometry to assess local in¯uence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local in¯uence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

We find sufficient conditions for principal toric bundles over compact Kähler manifolds to admit Calabi-Yau connections with torsion as well as conditions to admit strong Kähler connections with torsion. With the aid of a topological classification, we construct such geometry on (k − 1)(S 2 × S 4 )#k(S 3 × S 3 ) for all k ≥ 1.

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Abstract. We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in 3 complex dimensions.