Let $G$ be a Lie group which acts on a manifold $M$. We say that $M$ is $G$-oriented if $M$ is oriented and the $G$-action preserves the orientation of $M$.
I'm not sure what does it mean that "the $G$-action preserves the orientation of $M$" ?
I think that if $\omega$ is a volume form on M which determine its orientation, then we say that the $G$-action preserves the orientation of $M$ if $g.\omega$ is in the same class of $\omega$, for all $g \in G$. Am I right ?