Holonomic and semi-holonomic geometries (Q2765854)

In the framework of generalising the analysis on homogeneous spaces \(G/P\) to curved analogues of the flat model space \(G/P\), the paper under consideration introduces an alternative definition for a curved analogue of a homogeneous space. Indeed, holonomic and semi-holonomic geometries modelled on a homogeneous space \(G/P\) are introduced as reductions of the holonomic and semi-holonomic frame bundles, respectively, satisfying a straightforward generalisation of the partial differential equation characterising torsion-free linear connections. In the absence of isospin Morimoto had constructed a \(P\)-equivariant embedding of a Cartan geometry \({\mathcal G}\) on a manifold \(M\) into the infinite frame bundle. The main result of the current paper is a generalisation of this result, which provides a complete classification of Cartan geometries \({\mathcal G}\) on \(M\) modelled on \(G/P\) in terms of semi-holonomic geometries of sufficiently high order. Consequently, under a suitable regularity assumption on the model space \(G/P\) an equivalence of categories is established between Cartan geometries and semi-holonomic geometries modelled on \(G/P\).NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].