In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the *horizontal distribution*. One then has a Riemannian metric only on this distribution $\mathcal{H}$, and is interested in horizontal curves $\gamma$, i.e. those with $\dot{\gamma}(t) \in \mathcal{H}$.


If one has a principal bundle $P\longrightarrow M$ with connection $\omega$, this defines a horizontal distribution $\mathcal{H}$ on $P$. If one has a Riemannian metric on $M$, one can pullback this metric to $\mathcal{H}$. 


However, in the definition of a sub-Riemannian manifold, one usually has the requirement that $\mathcal{H}$ be *bracket generating*, i.e. the Lie hull of $\mathcal{H}$ is $TP$. In the principal bundle case, this is not always the case, e.g. when the connection $\omega$ is flat, because then $\mathcal{H}$ is integrable.


**My Questions are:**


1. Why does one pose this bracket-generating condition on $\mathcal{H}$ when doing sub-Riemannian geometry
2. Are there easy-to-formulate conditions on $\omega$ that tell my when the corresponding horizontal distribution satisfies the condition?


**Some Background:**


One can consider the horizontal Laplacian on $P$ defined by 
$$\Delta^{\mathrm{hor}} = -\sum\_{i=1}^n \bigl((e\_i^{\mathrm{hor}})^2 - (\nabla\_i e\_i)^{\mathrm{hor}}\bigr),$$
where $e\_i$ is an orthonormal basis of $M$, $\nabla$ is the Levi-Civita connection and ${\mathrm{hor}}$ denotes the horizontal lift. If $X\_t$ is a Brownian motion on $M$, then there is a horizontal lift $U\_t$ of $X\_t$ to $P$ (determined by the conditions $\pi U\_t = X\_t$ and that the Stratonovich integral $\int \omega(U\_t) \delta U\_t = 0$), and $\Delta^{\mathrm{hor}}$ is its generator.


I am interested in the transition functions (heat kernel) of this process, and in estimates on them. I learned that if $\mathcal{H}$ is bracket generating, then Hörmander's theorem applies and $\Delta^{\mathrm{hor}}$ is hypoelliptic. Otherwise, this might probably fail. 


So my third question is probably


3. What does this mean for the transition functions and the process $U\_t$?



Principal bundles and Subriemannian Geometry
Re question 2. The answer is provided by the Ambrose Singer theorem. You can see how it connects with your question in my book *A Tour of sub-Riemannian Geometry*. When the connection form is analytic then it is easy to state the Ambrose-Singer theorem. 
 The curvature of the connection, together with all its covariant derivatives, are forms with values in the Lie algebra of the structure group G of the principal bundle. Take the Lie algebra generated by all of these values and then
take the subgroup of the structure group having this Lie algebra. [This is the "holonomy group" of the bundle-with-connection.] The accessible set (points you can get to by horizontal paths) is a subbundle of the original bundle
whose fiber is this (possibly) smaller connected Lie group.


