Existence of abnormal minimizing geodesics in sub-Riemannian geometry
In sub-Riemannian geometry one studies manifolds equipped with a subbundle \(S\) of the tangent bundle \(TM\) and a Riemannian metric on \(S\). If \(S\) is fully nonholonomic, meaning that vector fields tangential to \(S\) together with their iterated Lie brackets generate all of \(TM\), then any two points of \(M\) can be joint by a curve tangential to \(S\), a so-called horizontal curve. This induces a distance function on \(M\), sometimes called Carnot-Caratheodory metric. It has been stated many times that shortest horizontal curves satisfy the corresponding Hamilton-Jacobi equations. In Riemannian geometry this is true: shortest curves satisfy the geodesic equation. But in sub-Riemannian geometry it turned out that there are shortest horizontal curves not satisfying the Hamilton-Jacobi equations. In the present paper a simple example for such an abnormal geodesic is constructed.
scientific article; zbMATH DE number 815611