The nonholonomic Riemann and Weyl tensors for flag manifolds

TL;DR

This paper introduces nonholonomic analogs of the Riemann and Weyl tensors, providing a framework to understand flatness obstructions in nonintegrable structures, with explicit calculations for flag manifolds and G(2)-structures.

Contribution

It defines nonholonomic Riemann and Weyl tensors and computes these tensors for flag varieties and G(2)-structures using advanced algebraic tools.

Findings

Defined nonholonomic Riemann and Weyl tensors.

Calculated tensors for flag varieties associated with simple Lie algebras.

Identified obstructions to flatness in nonholonomic structures.

Abstract

On any manifold, any non-degenerate symmetric 2-form (metric) and any skew-symmetric (differential) form W can be reduced to a canonical form at any point, but not in any neighborhood: the respective obstructions being the Riemannian tensor and dW. The obstructions to flatness (to reducibility to a canonical form) are well-known for any G-structure, not only for Riemannian or 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 tensors. With the help of Premet's theorems and a package SuperLie we calculate these tensors for the particular case of flag varieties associated with each maximal (and several other) parabolic subalgebra of each simple Lie algebra. We also compute obstructions to flatness of the G(2)-structure and its nonholonomic super counterpart.