The canonical isomorphism between the prolongation of the symbols of a nonlinear Lie equation and its attached linear Lie equation

J. Munoz, F. J. Muriel and J. Rodriguez


Abstract. Let $M$ be a smooth manifold and let us denote by $I^\ell(M)$ the fibre bundle of invertible $\ell$-jets from $M$ into $M$. For each point $\p^\ell$ the vertical tangent space, for the source projection $\alpha$, to $I^\ell(M)$ at $\p^\ell$ is isomorphic to the module of $\alpha^\ast C^\infty(M)$-derivations from $C^\infty(M\times M)$ into $C^\infty(M)/\m^{\ell+1}_{\alpha(\p^\ell)}$. Let $\mathcal R^\ell$ be a nonlinear Lie equation and $H^\ell$ its attached linear Lie equation; the composition of each vertical tangent vector to $\mathcal R^\ell$ at $\p^\ell$ with $(\p^\ell)^{-1}$, when it is meant as an $\R$-algebra isomorphism from $C^\infty(M)/\m^{\ell+1}_{\alpha(\p^\ell)}$ onto $C^\infty(M)/\m^{\ell+1}_{\beta(\p^\ell)}$, gives a natural isomorphism between the vertical fibre bundle $V\mathcal R^\ell$ and the pullback ${\mathcal R^\ell}\times_M H^\ell$ of $H^\ell$ by the projection $\beta$. From the interpretation of invertible jets and vertical vectors as homomorphisms and derivations, respectively, from $C^\infty(M\times M)$ into some Weil algebras follows that the $r$-prolongation preserves the translation which identifies the spaces $V_{\p^{\ell}}\mathcal R^{\ell}$ and $H^{\ell}_{\beta(\p^{\ell})}$, therefore $V_{\p^{\ell+r}}{\mathcal R}^{\ell+r}\approx H^{\ell+r}_{\beta(\p^{\ell+r})}$, without any assumptions about regularity of ${\mathcal R}^{\ell+r}$, and the symbol of ${\mathcal R}^{\ell+r}$ at $\p^{\ell+r}$ is isomorphic to the symbol of $H^{\ell+r}$ at $\beta(\p^{\ell+r})$.

AMSclassification. 58H05, 22E65

Keywords. Near points, invertible jets, symbol, Lie equation, pseudogroup