Dilatation structures in sub-riemannian geometry

Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of the properties of regular sub-riemannian manifolds by using the formalism of dilatation structures.

💡 Research Summary

The paper establishes a deep connection between the abstract theory of dilatation structures, introduced in arXiv:math/0608536, and the intrinsic geometry of regular sub‑Riemannian manifolds. A dilatation structure on a metric space (X,d) consists of a family of “zoom‑in” maps δ⁽ˣ⁾_ε : X → X, defined for every point x and every scale ε∈(0,1], which satisfy four axioms (A0–A4). These axioms encode continuity, identity at scale 1, compatibility of successive dilations, uniform contraction of distances, and a commutation rule for two‑step dilations. When the axioms hold, the space admits a well‑defined metric tangent cone at each point; this cone is a graded nilpotent Lie group (a Carnot group) equipped with a homogeneous distance. In this sense, dilatation structures provide a synthetic, coordinate‑free framework for metric differentiability.

The authors apply this framework to sub‑Riemannian geometry. A sub‑Riemannian manifold (M,Δ,g) consists of a smooth manifold M, a bracket‑generating distribution Δ, and a smoothly varying inner product g on Δ. The “regular” hypothesis means that the growth vector (the dimensions of the successive Lie brackets of Δ) is constant over M, so the step and the dimensions of each layer are globally fixed. Under this hypothesis, Bellaïche’s theorem guarantees the existence of normal frames: at each point x one can choose vector fields (X₁,…,Xₙ) that are adapted to the filtration and whose Lie brackets satisfy homogeneous scaling relations.

Using a normal frame, the authors define the dilations explicitly by  δ⁽ˣ⁾_ε(p) = expₓ( ε·logₓ(p) ), where logₓ and expₓ are the logarithmic and exponential maps constructed from the normal frame (they are only locally defined but sufficient for the analysis). Because the frame is homogeneous, the map ε·logₓ(p) scales each layer of the filtration by εⁱ, i.e. the i‑th layer is multiplied by εⁱ. This property is the key to checking the dilatation axioms.

The paper proceeds to verify each axiom:

• A0 (Domain and continuity) – δ⁽ˣ⁾_ε is defined on the whole manifold for ε∈(0,1] and varies continuously with ε and x, thanks to the smooth dependence of the normal frame.
• A1 (Identity at ε=1) – By construction δ⁽ˣ⁾₁ = idₘ.
• A2 (Compatibility of successive dilations) – The homogeneity of the normal frame yields δ⁽ˣ⁾ε ∘ δ⁽ˣ⁾{ε’} = δ⁽ˣ⁾_{εε’}, exactly as required.
• A3 (Uniform contraction of distances) – Using Hörmander’s condition and the Ball‑Box theorem, the authors show that the sub‑Riemannian distance d satisfies d(δ⁽ˣ⁾_ε(p),δ⁽ˣ⁾_ε(q)) ≤ C ε d(p,q) for a global constant C. This gives the uniform contraction property.
• A4 (Commutation of two‑step dilations) – The graded Lie algebra structure of the normal frame implies that the error between δ⁽ˣ⁾ε ∘ δ^{δ⁽ˣ⁾ε(y)}{ε’} and δ⁽ˣ⁾{εε’}(y) is o(ε). The authors provide a detailed estimate based on the Baker‑Campbell‑Hausdorff formula truncated at the step of the distribution.

Having verified A0–A4, the authors conclude that (M,d,δ) is a dilatation structure. The general theory of dilatation structures then yields several important consequences:

1. Metric Tangent Cones – At each point x, the rescaled spaces (M, d/ε, δ⁽ˣ⁾_ε) converge in the Gromov–Hausdorff sense to a Carnot group Gₓ equipped with a homogeneous distance. This recovers the classical result that the metric tangent cone of a regular sub‑Riemannian manifold is a Carnot group.
2. Metric Differentiability – Axiom A3 guarantees that the distance function is metrically differentiable; the differential is precisely the homogeneous norm on the tangent Carnot group. Consequently, length minimizers (geodesics) admit a first‑order approximation by straight lines in the tangent group.
3. Second‑Order Geometry – Axiom A4 provides a second‑order expansion of the distance, opening the way to define curvature‑type invariants in a purely metric fashion, analogous to the curvature tensors in Riemannian geometry.
4. Unified Proofs of Classical Results – Many known theorems (e.g., existence of privileged coordinates, Ball‑Box estimates, Popp’s volume, and the Mitchell theorem on tangent cones) become immediate corollaries of the dilatation‑structure framework, eliminating the need for ad‑hoc coordinate calculations.

Finally, the paper discusses the outlook for non‑regular (or non‑normal) sub‑Riemannian structures. Although a global normal frame may not exist, the authors suggest that local dilatation structures can still be built on neighborhoods where a partial normal frame is available, leading to a piecewise‑defined metric differential structure. This hints at a possible extension of the theory to more singular spaces, such as those with varying growth vectors or with abnormal geodesics.

In summary, the article proves that every regular sub‑Riemannian manifold naturally carries a dilatation structure, thereby providing an intrinsic, coordinate‑free description of its metric tangent cones, differentiability properties, and higher‑order geometry. The work not only unifies several classical results under a single abstract umbrella but also opens promising avenues for extending sub‑Riemannian analysis to more general metric spaces.