Relationship between smooth sections and group actions on manifold

Question

This question may be a bit naive, but I am relatively new to the study of differential topology. I am currently reading through John Lee's Smooth Manifolds. He devotes a section to the theory of integral curves of vector fields on manifolds, and the associated flow on the manifold. In the case of a complete vector field, the flow corresponds to a smooth $\mathbb{R}$-action on $M$. In later sections, vector fields on $M$ get fit into the more general context of smooth sections of bundles over $M$ (vector bundles, tensor bundles, etc.) In this language, certain smooth sections of the tangent bundle (ie vector fields) induce smooth $\mathbb{R}$ actions on $M$ (in the form of the flow). I am wondering if this phenomenon in anyway generalizes to smooth sections of other types of bundles. More explicitly, are there other contexts in which a smooth section of a bundle over $M$ induce some smooth Lie group action on $M$? Or is this phenomenon specific to vector fields? Thank you in advance!

Answer 1

Among bundles over $M$, $TM$ is somewhat special in that vector fields correspond to differential equations on $M$, while there is no such correspondence for sections of other bundles.

There is, however, a generalization of the notion of a generator of a Lie group action. The space of vector fields $\mathfrak{X}M$ is a Lie algebra under Lie brackets, and a Lie algebra homomorphism $\varphi:\mathfrak{g}\to\mathfrak{X}M$ can, under certain conditions, give rise to a Lie group action $\theta:G\times M\to M$ where $G$ is a Lie group whose lie algebra is $\mathfrak{g}$. In this way, Lie group actions correspond not with sections of $TM$, but with Lie subalgebras of $\mathfrak{X}M$. In the case of an $\mathbb{R}$-action generated by a vector field $V$, the relevant lie algebra morphism is $t\mapsto tV$.