Abstract
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.
Highlights
A n-dimensional Almost Riemannian Structure (n-ARS for short) is a rank-varying sub-Riemannian structure that can be locally defined by a set of n smooth vector fields on a n-dimensional manifold, satisfying the Hormander condition
More precisely we prove that the following properties hold under generic conditions1 (G1) the dimension of (q) is larger than or equal to 2 and q ∈ M; 1for the precise definition of generic see Definition 7 (q) + [, ](q) = TqM, for every
When the n-ARS is trivializable, the problem of finding a curve minimizing the energy between two fixed points q0, q1 ∈ M is naturally formulated as the distributional optimal control problem with quadratic cost and fixed final time nT n q = uiXi(q), ui ∈ R, u2i (t) dt → min, q(0) = q0, q(T ) = q1
Summary
A n-dimensional Almost Riemannian Structure (n-ARS for short) is a rank-varying sub-Riemannian structure that can be locally defined by a set of n (and not by less than n) smooth vector fields on a n-dimensional manifold, satisfying the Hormander condition (see for instance [2, 12, 31, 39]). The simplest example of analytic sub-Riemannian structure for which there are abnormal minimisers and for which the “sum of the square” is not analytic hypoelliptic is provided by a 3-ARS, namely the Baouendi-Goulaouic example, defined by the following three vector fields: 0. For this structure, the trajectory (0, y0 + t, 0) is an abnormal minimizer and the Green functions of the operator. The nilpotent approximation at a type-2 point is the Heisenberg sub-Riemannian structure and is not a 3-ARS. Appendix B contains the explicit construction of the heat kernel for ∆L
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