  Let $U$ be a connected open subset of $\mathbb{R}^n$, and let
$X=(X_1,X_{2},\ldots,X_m)$ be a system of H\"{o}rmander vector fields defined
on $U$. This paper addresses sharp embedding results and geometric inequalities
in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where
$\Omega\subset\subset U$ is a general open bounded subset of $U$. By employing
Rothschild-Stein's lifting technique and saturation method, we prove the
representation formula for smooth functions with compact support in $\Omega$.
Combining this representation formula with weighted weak-$L^p$ estimates, we
derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where
the critical Sobolev exponent depends on the generalized M\'{e}tivier index. As
applications of these sharp Sobolev inequalities, we establish the
isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov
compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and
Moser-Trudinger inequality in the context of general H\"{o}rmander vector
fields.
