I'm working on something related to differential geometry though I'm definitely not a geometer myself and I have encountered with the following object. Let $G$ be a Lie group acting on a differential manifold $M$ and for each $x\in M$ put $$ D_x = \{ f : T_xM \rightarrow G : f(0_x) = 1 \text{ and } d_{0_x}f = 0 \}. $$ Consider now $D=\coprod_{x\in M} D_x$. It seems to me that $D$ has to be well-known. Have you guys seen this somewhere?
2026-04-01 16:46:57.1775062017
$1$-forms on $M$ with values on a Lie group $G$?
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