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foliated manifold; infinitesimal automorphism; higher order adapted frame bundle; canonical $1$-form
Let $(M,\mathcal{F})$ be a foliated $m+n$-dimensional manifold $M$ with $n$-dimensional foliation $\mathcal{F}$. Let $V$ be a finite dimensional vector space over $\mathbf{R}$. We describe all canonical (${\mathcal{F}}\mbox {\it ol}_{m,n}$-invariant) $V$-valued $1$-forms $\Theta \colon TP^r(M,{\mathcal{F}}) \rightarrow V$ on the $r$-th order adapted frame bundle $P^r(M,\mathcal{F})$ of $(M,\mathcal{F})$.
[1] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer Verlag, 1993. MR 1202431
[2] Wolak, R. A.: Geometric structures on foliated manifolds. Publications del Departamento de Geometria y Topologia, Universidad de Santiago de Compostella 76 (1989). MR 1040852 | Zbl 0838.53029
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