Proposition 12

Let $X$ be a Lie group and $Y$ a non-empty analytic manifold with a law of analytic left operation of $X$ on $Y$. For all $y \in Y$, let $\varphi(y)$ be the orbital mapping by $y$ and $X_y$ the stabilizer of $y$ in $X$. The following conditions are equivalent: (i) there exists $y \in Y$ such that $\varphi(y)$ is a surjective submersion; (i') for all $y \in Y$, $\varphi(y)$ is a surjective submersion; (ii) there exists $y \in Y$ such that $X_y$ is a Lie subgroup of $X$ and the canonical mapping of $X/X_y$ into $Y$ is a manifold isomorphism; (ii') for all $y \in Y$, $X_y$ is a Lie subgroup of $X$ and the canonical mapping of $X/X_y$ into $Y$ is a manifold isomorphism; (iii) the mapping $(x, y) \mapsto (y, xy)$ of $X \times Y$ into $Y \times Y$ is a surjective submersion.

As the canonical mapping of $X$ onto $X/X_y$ is a submersion, the equivalences (i) $\Leftrightarrow$ (ii), (i') $\Leftrightarrow$ (ii') are immediate. (i) $\Leftrightarrow$ (i') by Proposition 9 of no. 5. The equivalence (i') $\Leftrightarrow$ (iii) follows from formula (7) of no. 5.

Under the conditions of Proposition 12, $Y$ is called a (left) Lie homogeneous space of $X$. Right Lie homogeneous spaces of $X$ are defined analogously.