Abstract
Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
Highlights
Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces
The space (S, dS) is called a length space, and dS the induced distance defined by (M, dsR). (In the theory of length metric spaces, the induced distance dS is called intrinsic distance, emphasising that it depends uniquely on lengths of curves in S, see [BBI01].) We stress that the induced distance dS is not the restriction dsR|S×S of the sub-Riemannian distance to S
This paper studies necessary and sufficient conditions on the surface S for which the induced distance dS is finite. i.e., dS(x, y) < +∞ for all points x, y in S; this is equivalent to (S, dS) being a metric space
Summary
M is a smooth 3-dimensional manifold, (D, g) a smooth contact subRiemannian structure on M , and S an embedded surface of class C2. If the orientability hypotheses hold, an equivalent definition of the characteristic foliation is the partition of S into the orbits of a global characteristic vector field This is a generalised foliation, as the dimension of the leaves is not constant since the characteristic set is partitioned in singletons. Any point in S admits a neighbourhood U in M in which there exists an oriented orthonormal frame (X1, X2) for D|U , and a submersion f of class C2 for which S is a level set, i.e., S ∩ U = f −1(0) and df |U = 0 In such case, a vector V ∈ T M |U is in T S if and only if V f = 0; for a point p ∈ U ∩ S,. For a more general discussion on the size of the characteristic set, we refer to [Bal03] and references therein
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