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MiS Preprint

The nonholonomic Riemann and Weyl tensors for flag manifolds

Pavel Grozman and Dimitry A. Leites


On any manifold, any non-degenerate symmetric 2-form (metric) and any non-degenerate skew-symmetric differential form $\omega$ can be reduced to a canonical form at any point, but not in any neighborhood: the respective obstructions being 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 the manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, though of huge interest (e.g., in supergravity) were not known until recently, though particular cases were known for more than a century (e.g., any contact structure is "flat": it can always be reduced, locally, to a canonical form).

We give a general definition of the nonholonomic analogs of the Riemann and Weyl (conformally invariant) tensors in terms of Lie algebra cohomology and retell Premet's theorems describing them. With the help of Premet's theorems and a package SuperLie, we calculate the spaces of values of these tensors for the particular case of flag varieties associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases). We also compute obstructions to flatness of the $G(2)$-structure and its nonholonomic super counterpart.

MSC Codes:
17A70, 17B35
Lie algebra cohomology, Cartan prolongation, Riemann tensor, nonholonomic manifold

Related publications

2007 Repository Open Access
Pavel Grozman and D. A. Leites

The nonholonomic Riemann and Weyl tensors for flag manifolds

In: Theoretical and mathematical physics, 153 (2007) 2, pp. 1511-1538