Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetildeπ:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $π: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $π: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetildeπ:\widetilde P\to \widetilde{H}/\widetilde{K}$. If $\widetildeπ:\widetilde P\to \widetilde{H}/\widetilde{K}$ is flat $k$-reductive we introduce a sequence of generalized curvatures taking values in appropriate cohomology groups and prove that the vanishing of these curvatures are necessary and sufficient conditions for the solution of the corresponding generalized integrability problems.
We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian st…
We propose a new, infinite class of brackets generalizing the Frölicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the ope…
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.
The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of Riemannian metrics can obstruct the integrability of almost-complex structures. It is shown that certain special complex structures cannot coexist with non-flat constant curvature metrics, and a formal variational realization of these structures is provided. The approach followed here is direct, meaning that it bypasses the classical theory. The idea is to find obst…
