Intrinsic torsion varieties

Abstract

At each point of an even-dimensional oriented Riemannian manifold M , the set of compatible complex structures is parametrized by the coset space Zn = SO(2n)/U(n) that is in fact Hermitian symmetric. The fact that Zn is a complex manifold is particularly significant. For example, it leads to the definition of twistor spaces [3, 8], and invariant Hermitian structures on a parallelizable manifold are typically described by a complex subvariety of Zn [1]. The fact that Zn is a symmetric space may be less significant. One can easily conceive of other subgroups H and homogeneous spaces SO(n)/H with which to classify geometrical structures on a Riemannian manifold. However, amongst these, the coadjoint orbits appear to have a privileged role, since they are simultaneously complex and symplectic manifolds. The resulting theory retains some features of the almost-Hermitian case, which it is the goal of this essay to generalize when n = 3. In six real dimensions, the coset space Z3 is isomorphic to complex projective space , arising from the double covering of Lie groups SU(4) → SO(6). As a consequence, we consider geometrical structures associated to other coadjoint orbits for SU(4), or equivalently SO(6). In particular, we shall discuss those almost-product structures that arise from the Grassmannians