A survey on sub-Riemannian geometry (Q2779219)

Let \(M\) be a connected \(C^\infty\) manifold of dimension \(m\) and \(S\) a subbundle of \(TM\) with fiber dimension \(r\) \((0<r<m)\). In this note the authors assume that \(S\) satisfies: Hörmander's condition. \(\Gamma(s)\) generates \(\Gamma(TM)\) as a Lie algebra. Also they present the following definition: A sub-Riemannian metric \(G\) is a Riemannian fiber metric on \(S\). The term ``sub-Riemannian'' metric is due to R. S. Strichartz (1986). Sub-Riemannian metrics appear in various contexts. Open Problem: Define the ``covariant derivative'' or the ``curvature'' in sub-Riemannian geometry. In the second part, the following is a fundamental result on length minimizing curves. Theorem. For any \(x\in M\) there is a neighborhood \(U\) of \(x\) such that any \(y\in U\) can be joined to \(x\) by a \(C^\infty\) curve \(c\) with length \(L(c)= d(x,y)\). (See the authors references). Also the authors present the following theorem (R. S. Strichartz): If \(S\) satisfies the strong bracket generating hypothesis, then necessary and sufficient condition for a continuous curve? \(I\to M\) to be locally a length minimizing curve is that \(c\) is a geodesic. The third part is consacrated to the examples in which sub-Riemannian metrics appear: control theory, hypoelliptic operators, nilpotent Lie groups. The fourth part treats different points of view of the curvature. The note contains a remark: Recently the paper of Z. Ge (1992) has been published and it presents certain analogues of the Levi-Civita connection and its curvature for sub-Riemannian manifold.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].