Comparison theorems for conjugate points in sub-Riemannian geometry.

*(English)*Zbl 1344.53023Let \(M\) be a sub-Riemannian manifold with horizontal distribution \(\mathcal{D}\subset TM\) and sub-Riemannian metric \(\langle\cdot | \cdot \rangle\). Suppose \(\gamma : [0,T] \to M\) is a geodesic, that is, a horizontal curve (\(\dot{\gamma} \in \mathcal{D}\)) which locally minimizes the sub-Riemannian length. In this paper, the authors study the conjugate time of \(\gamma\), which corresponds to the first critical point of the sub-Riemannian exponential map. (In many, but not all, cases, the first conjugate time is also the first time at which the geodesic ceases to be a global length minimizer.)

In Riemannian geometry, the conjugate time can be bounded in terms of the sectional and/or Ricci curvature along \(\gamma\), and the bounds are attained in spaces of constant sectional curvature. In the sub-Riemannian setting, the same notions of curvature are not generally available. Thus, in the present paper, the authors study a particular notion of sub-Riemannian directional curvature, and an analogous set of sub-Riemannian Ricci curvatures. An important consideration is the algebraic structure of the tangent space along \(\gamma\) with respect to iterated Lie derivatives of horizontal vector fields (the so-called geodesic flag). The authors find it useful to represent this structure using a Young diagram \(D\), as introduced in [I. Zelenko and C. Li, Differ. Geom. Appl. 27, No. 6, 723–742 (2009; Zbl 1177.53020)], whose levels index the set of Ricci curvatures.

Instead of spaces of constant sectional curvature, the “model spaces” for comparison here are linear-quadratic optimal control problems (LQ problems, for short). In such a problem, one is given matrices \(A^{n \times n}\), \(B^{n \times k}\), \(Q^{n \times n}\), points \(x_0\), \(x_1 \in \mathbb R^n\), and a time \(t>0\). A path \(x:[0,t] \to \mathbb R^n\) is admissible for the LQ problem if it satisfies \(x(0) =x_0\), \(x(t) = x_1\), and there exists a control path \(u : [0,t] \to \mathbb R^k\) so that \(\dot{x} = Ax + Bu\). (Note this is a linear constraint.) Then one seeks to minimize the quadratic functional \(\phi_t(u) = \frac{1}{2} \int_0^t (u^\ast u - x^\ast Q x)\,dt\) over all admissible paths. Solutions of an associated Hamiltonian system are local minimizers of \(\phi_t\), but have a conjugate time at which they fail to globally minimize.

The first main result is a comparison theorem, stating that if the directional curvatures along a geodesic \(\gamma\) are bounded above (respectively, below) by a quadratic form \(Q\), then the conjugate time is bounded below (respectively, above) by the conjugate time of an LQ problem in which the functional \(\phi_t\) is defined by \(Q\), and the constraint matrices \(A\), \(B\) are defined from the Young diagram \(D\). The authors also prove an average comparison theorem, in which one needs lower bounds on the sub-Riemannian Ricci curvatures corresponding to each level of the Young diagram \(D\). Again, the conclusion is that the conjugate time is bounded above by that of a certain natural LQ problem. As a consequence, the authors derive a Bonnet-Myers-type theorem, giving compactness of the manifold \(M\) under a positivity-like condition on the Ricci curvatures.

An important tool in the proof is to associate the space of Jacobi fields along a curve in the cotangent space with a Riccati equation whose blowup time gives the conjugate time of a geodesic.

The authors conclude the paper by discussing their results in the special case of 3-dimensional unimodular Lie groups.

In Riemannian geometry, the conjugate time can be bounded in terms of the sectional and/or Ricci curvature along \(\gamma\), and the bounds are attained in spaces of constant sectional curvature. In the sub-Riemannian setting, the same notions of curvature are not generally available. Thus, in the present paper, the authors study a particular notion of sub-Riemannian directional curvature, and an analogous set of sub-Riemannian Ricci curvatures. An important consideration is the algebraic structure of the tangent space along \(\gamma\) with respect to iterated Lie derivatives of horizontal vector fields (the so-called geodesic flag). The authors find it useful to represent this structure using a Young diagram \(D\), as introduced in [I. Zelenko and C. Li, Differ. Geom. Appl. 27, No. 6, 723–742 (2009; Zbl 1177.53020)], whose levels index the set of Ricci curvatures.

Instead of spaces of constant sectional curvature, the “model spaces” for comparison here are linear-quadratic optimal control problems (LQ problems, for short). In such a problem, one is given matrices \(A^{n \times n}\), \(B^{n \times k}\), \(Q^{n \times n}\), points \(x_0\), \(x_1 \in \mathbb R^n\), and a time \(t>0\). A path \(x:[0,t] \to \mathbb R^n\) is admissible for the LQ problem if it satisfies \(x(0) =x_0\), \(x(t) = x_1\), and there exists a control path \(u : [0,t] \to \mathbb R^k\) so that \(\dot{x} = Ax + Bu\). (Note this is a linear constraint.) Then one seeks to minimize the quadratic functional \(\phi_t(u) = \frac{1}{2} \int_0^t (u^\ast u - x^\ast Q x)\,dt\) over all admissible paths. Solutions of an associated Hamiltonian system are local minimizers of \(\phi_t\), but have a conjugate time at which they fail to globally minimize.

The first main result is a comparison theorem, stating that if the directional curvatures along a geodesic \(\gamma\) are bounded above (respectively, below) by a quadratic form \(Q\), then the conjugate time is bounded below (respectively, above) by the conjugate time of an LQ problem in which the functional \(\phi_t\) is defined by \(Q\), and the constraint matrices \(A\), \(B\) are defined from the Young diagram \(D\). The authors also prove an average comparison theorem, in which one needs lower bounds on the sub-Riemannian Ricci curvatures corresponding to each level of the Young diagram \(D\). Again, the conclusion is that the conjugate time is bounded above by that of a certain natural LQ problem. As a consequence, the authors derive a Bonnet-Myers-type theorem, giving compactness of the manifold \(M\) under a positivity-like condition on the Ricci curvatures.

An important tool in the proof is to associate the space of Jacobi fields along a curve in the cotangent space with a Riccati equation whose blowup time gives the conjugate time of a geodesic.

The authors conclude the paper by discussing their results in the special case of 3-dimensional unimodular Lie groups.

Reviewer: Nathaniel Eldredge (Greeley)

##### MSC:

53C17 | Sub-Riemannian geometry |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

53C22 | Geodesics in global differential geometry |

49N10 | Linear-quadratic optimal control problems |

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\textit{D. Barilari} and \textit{L. Rizzi}, ESAIM, Control Optim. Calc. Var. 22, No. 2, 439--472 (2016; Zbl 1344.53023)

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