The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: $\omega =dz-\frac{y^2}{2}dx$, q=(x,y,z) and g is a metric on D which can be taken in the normal form: $g=a\left(q\right)d{x}^2+c\left(q\right)d{y}^2$, a=1+yF(q), c=1+G(q), $G_{{|}_{x=y=0}}=0$. In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case....

We study minimal surfaces in sub-riemannian manifolds with sub-riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail $2$-dimensional surfaces in contact manifolds of dimension $3$. We show that in this case minimal surfaces are projections of a special class of $2$-dimensional surfaces...

We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces...