Sub-Riemannian geometry
A sub-Riemannian or singular Riemannian geometry is given by a smoothly varying positive definite quadratic form defined only on a subbundle \(S\) of the tangent bundle \(TM\) of a differentiable manifold, \(S\) being bracket-generating, that is sections of \(S\) together with their Lie brackets generate the \(C^{\infty}(M)\)-module \(V(M)\) of vector fields. Such a metric is also called aCarnot-Carathéodory metric. The paper gives a fairly self-contained description of the known results concerning length minimizing curves and geodesics, the exponential map, completeness and isometries, the latter allowing the investigation of sub-Riemannian symmetric spaces, which, in the case of three dimensions, are completely classified up to local isometry. For piecewise smooth curves tangent to the subbundle \(S\) there is a natural notion of length. If \(M\) is connected, by a theorem of Chow (1939) it is possible to connect any two points of \(M\) by such a curve, giving \(M\) the structure of a metric space. The sub-Riemannian metric gives a natural positive semidefinite quadratic form on covectors and therefore a quadratic function on the cotangent bundle, the Hamiltonian vector field of which replaces the geodesic spray of a Riemannian manifold. The projections of the integrals of this vectorfield are the geodesics of the sub-Riemannian structure. The main results concerning geodesics are: Every length-minimizing curve is a geodesic and locally every geodesic is length-minimizing, the latter result as some others only being derived under the so-called strong bracket generating hypothesis, that is the sections of S and the brackets of such sections with any non-zero section of \(S\) generate \(V(M)\). The proof requires a careful study of the exponential map which can be defined in a natural way as a map from \(T^*M\) to \(M\). In contrast to the Riemannian situation the exponential map never is a diffeomorphism at the origin. Another basic result is the analogue of the Hopf-Rinow theorem, the completeness of \(M\) being proved under the mentioned hypothesis. Possibly for some results this hypothesis can be dropped, but, as the exponential map looses some of its properties, other methods will be required. Corollary 6.2 and the other results depending on it have to be modified because the original proof is not correct. Also some other corrections are indicated.
scientific article; zbMATH DE number 3986104