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4
votes
1answer
215 views

Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?

In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". Can someone explain what are the major ...
3
votes
0answers
73 views

The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
3
votes
2answers
154 views

Converse to Chow's theorem in sub-riemannian geometry

Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of ...
3
votes
2answers
201 views

Ball-Box Theorem and Sequence of Distributions

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 ...
1
vote
3answers
399 views

Open problems in sub-Riemannian geometry

What are some open problems in sub-Riemannian geometry? I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.