Nonholonomic Riemann and Weyl tensors for flag manifolds

Abstract

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form omega can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and d omega. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically flat: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet's theorems describing these cohomologies. Using Premet's theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.