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Comparison theorems for conjugate points in sub-Riemannian geometry. (English) Zbl 1344.53023
Let $$M$$ 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.

##### 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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