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 {\tt 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.
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