A generalized integrability problem for G-Structures

TL;DR

This paper investigates the conditions under which a given G-structure on a manifold can be realized as a substructure induced from a larger homogeneous G-structure, using generalized curvature invariants.

Contribution

It introduces a framework for analyzing generalized integrability of G-structures via a sequence of curvature invariants in cohomology groups, extending classical integrability criteria.

Findings

Vanishing of generalized curvatures characterizes integrability.

Provides necessary and sufficient conditions for G-structure realizability.

Extends classical integrability theory to broader geometric contexts.

Abstract

Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetilde\pi:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $\pi: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $\pi: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$. If $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$ 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.