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Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?

Denote by $G$ the 1-jets of maps in $\Gamma$: $G := \lbrace j^1_p f \, \vert \, f \in \Gamma, p \in \mathrm{dom}(f) \rbrace$. Is $G \subset J^1(M,M)$ a submanifold of the first jet manifold? …