Almost-Riemannian Geometry of the Two-Sphere

Consider the two following vector fields on $S^2$:
\begin{equation*} f_1(x)= x\times e_2, \qquad f_2(x) = x\times\sqrt{1-a^2}e_1, \qquad x\in\mathbb{R}^3, \qquad |x| = 1, \end{equation*}
where $e_i$, $i=1,2,3$ is the standard basis of $\mathbb{R}^3$ and $a\in(0,1)$ is a parameter. These vector fields correspond to rotations around axis $OX_2$ and $OX_1$.
Vector fields $f_1$ and $f_2$ span a non-constant rank distribution $\Delta$:
\begin{equation*} \Delta_x = \operatorname{span}\{f_1(x),f_2(x)\}. \end{equation*}
It's easy to see that $\operatorname{rank} \Delta_x = 2$ almost everywhere, except for the equator, where $f_1$ and $f_2$ are collinear. The equator $\{x\in S^2: x_3 = 0\}$ is called the singular set. Nevertheless any two points can be joined by a horizontal curve, which follows from the fact that
\begin{equation*} \Delta_x + [\Delta,\Delta]_x = T_xS^2. \end{equation*}

Assume that there is a scalar product $g(\cdot,\cdot)$ on $\Delta$ for which the two vector fields $f_1$ and $f_2$ are othonormal:
\begin{equation*} g(f_i,f_j)=\delta_{ij}, \qquad i,j=1,2. \end{equation*}
A triple $(S^2, \Delta, g)$ is called an almost-Riemannian sphere. In fact, everywhere except the singular set metric $g$ is just a Riemannian metric on the sphere.
In the talk the problem of finding minimal curves of this structure will be discussed. This problem can be formulated as an optimal control problem:
\begin{gather*} \dot{x} = u_1f_1(x) + u_2f_2(x), \\[5pt] x,\omega\in \mathbb{R}^3, \qquad |x|=1, \\[5pt] (u_1,u_2)\in\mathbb{R}^2, \qquad a\in(0,1), \\[5pt] x(0) = \gamma_0, \qquad x(T)=x_T, \\[5pt] \int_0^T \sqrt{u_1^2+u_2^2}\,dt\rightarrow\min. \end{gather*}
We'll give a full parameterization of the geodesics and show how this problem is connected with the sub-Riemannian problems on SO(3). We'll also give description of Maxwell sets and bounds on the cut time.