Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:


Let $\{X\_j\} \in Vect(M)$ be a finite "basis" of some integrable subbundle of $TM$ (meaning that their locally linearly independent and closed under lie bracket). We have $Span\{X\_j\} \cong \mathfrak{g}$ for some finite dimensional lie algebra. The exponential map $\varphi : \mathfrak{g} \to G$ gives a group equipped with a natural action $\rho: G \to Diff(M)$. Denoting the flow of $X$ by $\varphi\_t^X$ the action looks like:


$$\rho : g=e^{X\_1}e^{X\_2}\dots e^{X\_n} \mapsto \varphi\_1^{X\_1}\circ e\_1^{X\_2}\circ\dots \circ\varphi\_1^{X\_n}(-)$$



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> **Question 1: Is this map well defined?**
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In any case if $\mathfrak{g}$ is abelian the action is well define and we get a product of circles and lines inside $Diff(M)$. For every point $x \in M$ the action of the torus part will carve an embedded submanifold of $M$ and the action of the euclidean part will carve an immersed submanifold. 



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> **Question 2: Is there anything more substantial to say here? When will the
>  action of the torus part yield an actual torus?**
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> **Question 3: Does this construction give a $G/\ker\rho$-fiber bundle?** (does it help if $G$ is compact?). 
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Vector fields, diffeomorphism subgroups and lie group actions
Every orbit of a torus is a torus, since every orbit of a Lie group action is a homogeneous space of the Lie group.


