Shortest paths for sub-Riemannian metrics on rank-two distributions.

*(English)*Zbl 0843.53038
Mem. Am. Math. Soc. 564, 104 p. (1995).

A sub-Riemannian metric on a smooth manifold \(M\) is defined by a smooth bracket generating distribution \(E\) on \(M\) and a smooth Riemannian metric on \(E\). The book gives an excellent self-contained introduction into the analysis of locally minimum arcs of such metrics and provides a fairly complete description of these arcs in the case that \(E\) is two-dimensional and satisfies some mild regularity conditions. Many of the results are new and correct various false claims and incomplete proofs in the literature. The results are illustrated by various examples.

The point of view of the authors is that of optimal control theory in a Hamiltonian setting. The techniques and formalisms of control theory are used throughout. This also includes the introduction of a systematic terminology for the various types of curves considered which, however, is not very imaginative and might lead to confusions.

The Riemannian metric on \(E\) induces naturally a smooth quadratic form on the cotangent bundle \(T^*M\) of \(M\) and hence an energy functional \(H\). The projection to \(M\) of an integral curve \(\gamma\) of the Hamiltonian vector field of \(H\) with \(H(\gamma )\neq 0\) is a solution of the Hamilton-Jacobi equation for the sub-Riemannian metric and is called abnormal extremal by the authors. A control theoretic proof is given that these curves are locally minimizing. But there may be more local minimizers: In the case that \(E\) is 2-dimensional and satisfies some regularity conditions, the submanifold \(N\) of \(T^* M\) of functionals which annihilate \(E+ [E,E ]\) but not \([E, [E,E ]]\) admits a 1-dimensional foliation such that each leaf of the foliation projects to a locally minimizing curve \(\varphi\) which is not a normal extremal. All of these curves are moreover \(C^1\)-rigid in the sense of Bryant-Hsu.

The point of view of the authors is that of optimal control theory in a Hamiltonian setting. The techniques and formalisms of control theory are used throughout. This also includes the introduction of a systematic terminology for the various types of curves considered which, however, is not very imaginative and might lead to confusions.

The Riemannian metric on \(E\) induces naturally a smooth quadratic form on the cotangent bundle \(T^*M\) of \(M\) and hence an energy functional \(H\). The projection to \(M\) of an integral curve \(\gamma\) of the Hamiltonian vector field of \(H\) with \(H(\gamma )\neq 0\) is a solution of the Hamilton-Jacobi equation for the sub-Riemannian metric and is called abnormal extremal by the authors. A control theoretic proof is given that these curves are locally minimizing. But there may be more local minimizers: In the case that \(E\) is 2-dimensional and satisfies some regularity conditions, the submanifold \(N\) of \(T^* M\) of functionals which annihilate \(E+ [E,E ]\) but not \([E, [E,E ]]\) admits a 1-dimensional foliation such that each leaf of the foliation projects to a locally minimizing curve \(\varphi\) which is not a normal extremal. All of these curves are moreover \(C^1\)-rigid in the sense of Bryant-Hsu.

Reviewer: U.Hamenstädt (Bonn)