Transverse Lie jets and holomorphic geometric objects on transverse bundles (Q2629434)

Given a foliated manifold \((M,\mathcal{F})\) locally isomorphic to \(\mathbb{R}^n\times \mathbb{R}^m\) (with copies of \(\mathbb{R}^m\) being the leaves) and a Weil algebra \(\mathbb{A}\), one constructs a \textit{bundle of transverse \(\mathbb{A}\)-velocities} \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\) locally modeled on \(\mathbb{A}^n\oplus\mathbb{R}^m\). Roughly speaking to build \(\mathrm{T}^\mathbb{A}_{\mathrm{tr}}M\) we consider the full bundle of \(\mathbb{A}\)-velocities \(\mathrm{T}^{\mathbb{A}} M\) on \(M\) and forget the parts of \(\mathbb{A}\)-velocities which are tangent to the leaves. \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}} M\) is naturally a bundle over \(M\) and an \(\mathbb{A}\)-manifold. It turns out that \(\mathbb{A}\)-automorphisms of this bundle (i.e., \(\mathbb{A}\)-diffeomorphisms which are bundle maps and leave the base invariant) are in 1-1 correspondence with sections \(\alpha:M\rightarrow\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}} M\). On \(M\) one considers a \textit{bundle of foliated \(r\)-frames} \(P^r_{\mathrm{fol}}M\), i.e., \(r\)-jets of germs of foliation isomorphisms \((\mathbb{R}^n\times\mathbb{R}^m,0)\rightarrow M\). This is naturally a \(G\)-bundle, with \(G\) being the group of \(r\)-jets of germs of foliation automorphisms \((\mathbb{R}^n\times\mathbb{R}^m,0)\rightarrow (\mathbb{R}^n\times\mathbb{R}^m,0)\). Given a \(G\)-space \(\beta:G\times F\rightarrow F\) one considers a \textit{field of geometric objects of type \((F,\beta)\) on \(M\)} understood as a \(G\)-equivariant map \(\lambda:P^r_{\mathrm{fol}}M\rightarrow F\). Additionally we want \(F\) to be \(G\)-fibrated over a \(G\)-space \(\overline{F}\). If \(\lambda\) projects to a \(G\)-equivariant map \(\overline{\lambda}:P^r_{\mathrm{tr}}M\rightarrow \overline{F}\) (here \(P^r_{\mathrm{tr}}M\) are foliated transverse \(r\)-frames on \(M\)), we call \(\lambda\) a \textit{field of transverse geometric objects of type \((F,\beta)\) on \(M\)}. By Theorem 1, every such a field \(\lambda\) can be naturally \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}\)-lifted to an \textit{\(\mathbb{A}\)-field of geometric objects of type \((\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}F,\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}\beta)\) on \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}} M\)}, i.e., a \(G^{\mathbb{A}}\)-equivariant map \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}\lambda:P^r(\mathbb{A})\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\rightarrow \mathrm{T}^{\mathbb{A}}_{\mathrm{tr}} F\). Here \(P^r(\mathbb{A})\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\) is a bundle of \(r\)-jets of germs of \(\mathbb{A}\)-diffeomorphisms \((\mathbb{A}^n\oplus\mathbb{R}^m,0)\rightarrow \mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\), being a \(G^{\mathbb{A}}\)-principal bundle, where \(G^{\mathbb{A}}\) is a group of \(r\)-jets of germs of \(\mathbb{A}\)-automorphisms \((\mathbb{A}^n\oplus\mathbb{R}^m,0)\rightarrow(\mathbb{A}^n\oplus\mathbb{R}^m,0)\). The authors use the (in my opinion misleading) term \textit{bundle of holomorphic \(r\)-frames on \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\)} to describe \(P^r(\mathbb{A})\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\). On the other hand, \(G^{\mathbb{A}}\)-equivariant maps \(\Lambda:P^r(\mathbb{A})\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\rightarrow \mathrm{T}^{\mathbb{A}}_{\mathrm{tr}} F\) can be considered on their own right. The main result of the paper is Theorem 2 which provides a characterization of these \(\mathbb{A}\)-fields \(\Lambda\) of geometric objects of type \((\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}F,\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}\beta)\) on \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\) which are \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}\)-lifts of some field \(\lambda\) of transverse geometric objects of type \((F,\beta)\) on \(M\) up to an \(\mathbb{A}\)-bundle automorphism of \(\mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\) induced by some section \(\alpha:M\rightarrow \mathrm{T}^{\mathbb{A}}_{\mathrm{tr}}M\). This result is then illustrated with an example of a linear connection on \(M\).