Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on some domain $D(x)$ around x. Moreover let $(e^k,g^k)$ converge to another distribution $(e,g)$

i). The ball-box theorem states that there exists constants $C_k,c_k$ and $r$ s.t

$exp_x(Box(c_kr)) \subset B^k_{sR}(r) \subset exp_x(Box(C_kr))$

where $B^k_{sR}(r)$ is the subriemannian ball of $k^{th}$ distribution. Now my first question is how does these constants depend on the distribution. From the proofs I have studies it seemes upper constant can be chosen to only depend on behaviour of $e^k,g^k$. The lower constant however depends on the map which exponentiates $e^k,g^k,[e^k,g^k]$. Thus it seems to me it would be uniform if for instance $(f^k,[e^k,g^k])$ is uniformly bigger than some constant and decreasing if this quantity goes to zero? However the magn. of $c_k$ depends on the implicit inner workings of inverse function theorem (in the proofs I have seen).

ii). Looking back at the ball box theorem if $(f^k,[e^k,g^k])$ goes to 0 then the boxes (or parallelopipeds) $exp_x(Box(c_kr))$ are becoming increasingly thin in the $f^k$ direction therefore the subRiemannian balls are converging to a 2 dimensional hypersurface? I am not sure how this reasoning could be true since it suggest that also the orbit of this distribution for time less than r also converges to a plane. But even if $(f^k,[e^k,g^k])$ goes to 0, further Lie brackets may be non-zero thus contributing to the orbit. Moreover even the limit distribution e,g may be non-involutive since the Lie brackets might not be zero. These are the reasons why this reasoning seems faulty to me.

I am new to sR geometry so far have looked at book/notes of ivanov, agrachev and gromov. I would also be grateful if you have seen and could direct me to any resources where sequence of distributions are handled or studied.

Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on some domain $D(x)$ around x. Moreover let $(e^k,g^k)$ converge to another distribution $(e,g)$

i). The ball-box theorem states that there exists constants $C_k,c_k$ and $r$ s.t

$exp_x(Box(c_kr)) \subset B^k_{sR}(r) \subset exp_x(Box(C_kr))$

where $B^k_{sR}(r)$ is the subriemannian ball of $k^{th}$ distribution. Now my first question is how does these constants depend on the distribution. From the proofs I have studies it seemes upper constant can be chosen to only depend on behaviour of $e^k,g^k$. The lower constant however depends on the map which exponentiates $e^k,g^k,[e^k,g^k]$. Thus it seems to me it would be uniform if for instance $(f^k,[e^k,g^k])$ is uniformly bigger than some constant and decreasing if this quantity goes to zero? However the magn. of $c_k$ depends on the implicit inner workings of inverse function theorem (in the proofs I have seen).

ii). Looking back at the ball box theorem if $(f^k,[e^k,g^k])$ goes to 0 then the boxes (or parallelopipeds) $exp_x(Box(c_kr))$ are becoming increasingly thin in the $f^k$ direction therefore the subRiemannian balls are converging to a 2 dimensional hypersurface? I am not sure how this reasoning could be true since it suggest that also the orbit of this distribution for time less than r also converges to a plane. But even if $(f^k,[e^k,g^k])$ goes to 0, further Lie brackets may be non-zero thus contributing to the orbit. Moreover even the limit distribution e,g may be non-involutive since the Lie brackets might not converge. These are the reasons why this reasoning seems faulty to me.

I am new to sR geometry so far have looked at book/notes of ivanov, agrachev and gromov. I would also be grateful if you have seen and could direct me to any resources where sequence of distributions are handled or studied.