Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?


Specifically the case of a right invariant affine distribution: $D\_{U} = \{ aU + \lambda bU | a,b \in \mathfrak{su}(n), \lambda \in \mathbb{R} \}$ on $SU(n)$ is of interest. I am aware of the work of Bryant and Hsu on this issue in the case of a rank 2 distribution and I'm interested in any thing similar in the affine case.



Singular curves of affine distributions on a Lie group
**Corrected Answer**


This is a special case of a more general fact: Suppose that one has two linearly independent vector fields $A$ and $B$ on a manifold $M^n$ and that one wants to study the regularity of the curves $\gamma:[a,b]\to M$ that satisfy
$$
\gamma'(t) = A\bigl(\gamma(t)\bigr) + u(t)\,B\bigl(\gamma(t)\bigr)
$$
for some function $u(t)$. In other words, the 'control curves' are defined to be the curves $\gamma:[a,b]\to M$ whose velocity vectors lie in the affine subbundle
$$
Z = \bigl\{\ A(p) + u B(p)\ \ |\ \ p\in M\ \text{and}\ u\in\mathbb{R}\ \bigr\}\subset TM.
$$
This can be reduced to the case of studying the regularity of curves tangent to a linear $2$-plane field as follows: Let $\pi:Z\to M$ be the basepoint projection. For each $z \in Z$, define the subspace $D\_z\subset T\_zZ$ of dimension $2$ by
$$
D\_z = \bigl(\pi'(z)\bigr)^{-1}\bigl(\,\mathbb{R}\cdot z\,\bigr).
$$
(Since $\pi$ is a submersion with $1$-dimensional fibers, $D\_z$ is a $2$-dimensional subspace of $T\_zZ$.)


Any differentiable $Z$-curve $\gamma:[a,b]\to M$ (i.e., one whose velocity vectors lie in $Z$) has the property that its tangential lift $\gamma':[a,b]\to Z$ is a $D$-curve. Conversely any $D$-curve $\phi:[a,b]\to Z$ that is transverse to the fibers of $\pi$ has the property that it can be reparametrized so as to be the tangential lift of a (differentiable) $Z$-curve.


It is not hard to show that a $Z$-curve is regular if and only if its tangential lift is regular as a $D$-curve. Hence, the well-known criteria for regularity of $D$-curves (see Proposition 1 of the Bryant-Hsu paper *Rigidity of integral curves of rank $2$ distributions*, Inventiones Math. **114** (1993), 435–461) translates directly to give criteria for regularity of $Z$-curves.


In the case in which $M$ is a Lie group $G$ (compact or not) and $A$ and $B$ are right-invariant vector fields on $G$, the problem of constructing all of the non-regular $Z$-curves then reduces to a purely Lie-algebraic problem. 


(Note that the result will depend on the particular vector fields $A$ and $B$. For example, if $A$ and $B$ do not generate the full Lie algebra of $G$, then the endpoint mapping will *never* be surjective, so, by the OP's definition, *all* of the $Z$-curves will be singular in this case.) 


