Sub-Riemannian Transport

• Sub-Riemannian transport is a framework for optimal mass transfer on manifolds using horizontal trajectories constrained by bracket-generating distributions.
• It adapts static (Kantorovich) and dynamic (Benamou–Brenier) formulations to accommodate nonholonomic constraints, abnormal minimizers, and anisotropic geodesic behavior.
• The theory underpins applications in control theory, stochastic processes, and physical systems, with concrete examples including the Heisenberg group and ocean neutral transport.

Sub-Riemannian transport refers to the theory and analysis of optimal mass transportation where the geometry of the underlying space is sub-Riemannian rather than Riemannian. In contrast with classical (Euclidean or Riemannian) optimal transport, sub-Riemannian transport involves additional constraints on admissible curves: only horizontal trajectories (i.e., those tangent to a prescribed distribution that is bracket-generating but not full rank everywhere) are permitted. This leads to profound differences in the structure of geodesics, transport maps, regularity, and metric inequalities, and increasingly finds applications across control theory, stochastic processes, thermodynamics, data-driven modeling, and physical systems with nonholonomic constraints (Abdelgalil et al., 2024).

1. Sub-Riemannian Geometry and Foundations of Optimal Transport

A sub-Riemannian manifold $(M, \Delta, g)$ consists of a smooth connected manifold $M$, a smooth bracket-generating distribution $\Delta \subset TM$, and a smooth inner product $g$ defined on $\Delta$. A Lipschitz curve $\gamma: [0,1] \to M$ is horizontal iff $\dot\gamma(t) \in\Delta_{\gamma(t)}$ a.e.; its length is $l(\gamma) = \int_0^1 \|\dot{\gamma}(t)\|_g \,dt$. The sub-Riemannian (Carnot–Carathéodory) distance is

$$d_{SR}(x,y) = \inf \{ l(\gamma) : \gamma(0)=x,\,\gamma(1)=y,\,\dot{\gamma} \in \Delta \}.$$

The bracket-generating (Hörmander) condition ensures that $d_{SR}$ is finite and defines the topology of $M$0.

Optimal mass transport (OMT) in this setting involves finding, for $M$1, a minimizer to

$M$2

with $M$3. The Monge and Kantorovich formulations (relaxing to transport plans) generalize from Riemannian to sub-Riemannian geometry, but with new subtleties regarding geodesics, regularity, and the existence of abnormal minimizers (Badreddine, 2017, Barilari et al., 2017, Citti et al., 28 Jul 2025).

2. Geodesics, Horizontal Distributions, and Sub-Riemannian Distance

Sub-Riemannian geodesics correspond to curves solving Hamiltonian equations governed by the sub-Riemannian Hamiltonian $M$4. Normal geodesics are projections of flows with $M$5; abnormal geodesics arise as critical points of the endpoint map and may have $M$6. The presence or absence of nontrivial abnormal generators ('ideal' structures) is decisive for the uniqueness and regularity of optimal maps: if abnormal minimizers are absent, the squared distance $M$7 is semiconvex away from the cut locus and standard regularity results (e.g., uniqueness of optimal transport, Jacobian estimates) hold (Barilari et al., 2017).

For example, on the Heisenberg group $M$8 (as a step-2 Carnot group), the horizontal distribution is spanned by $M$9 left-invariant vector fields, and geodesics exhibit characteristic sub-Riemannian behavior: motion is 'fast' along horizontal directions and 'slow' in the directions generated via their brackets. This anisotropy is formalized in the asymptotics of the Carnot–Carathéodory distance, which locally scales linearly in horizontal and sub-linearly in “vertical” directions (Circelli et al., 2023, Chatelain et al., 3 Nov 2025).

3. Sub-Riemannian Optimal Transport Theory: Formulations and Structure

Both static (Kantorovich) and dynamic (Benamou--Brenier) formulations admit sub-Riemannian generalizations. The Benamou--Brenier framework seeks curves $\Delta \subset TM$0 subject to a sub-Riemannian continuity equation

$\Delta \subset TM$1

with $\Delta \subset TM$2 and cost $\Delta \subset TM$3. The equivalence between dynamic and static formulations holds under completeness and absence of abnormal minimizers: every optimal plan arises from displacement interpolation along normal geodesics (Citti et al., 28 Jul 2025). The geodesic equation in probability space inherits sub-Riemannian structure, and the tangent space at a density $\Delta \subset TM$4 is characterized by the sub-elliptic continuity equation $\Delta \subset TM$5, with metric

$\Delta \subset TM$6

mirroring the Otto formalism but using the horizontal Laplacian (Feng et al., 2019).

Uniqueness and regularity of Monge minimizers can be proven in settings where the singular set (points joined via singular minimizers) is negligible, leveraging the C$\Delta \subset TM$7-semiconvexity of Kantorovich potentials and geometric growth properties of the distribution (Badreddine, 2017).

4. Holonomy, Mixing, and Labeled-Particle Sub-Riemannian OMT

A major development is the analysis of 'labeled-particle' optimal transport, where the sub-Riemannian structure arises in the space of covariance factors in the finite-dimensional Gaussian case. The configuration manifold is $\Delta \subset TM$8, with the projection $\Delta \subset TM$9 a principal $g$0-bundle. The horizontal distribution is determined by the Ehresmann connection: $g$1 The corresponding sub-Riemannian metric is $g$2 with $g$3, $g$4. Normal geodesics satisfy a Hamiltonian system with no abnormal extremals.

A novel phenomenon emerges: along closed loops in the base (covariances), horizontal lifts yield a holonomy $g$5 which measures the 'mixing' or permutation of labels. In classical (Riemannian) OMT, such mixing is generic and unconstrained, but in the sub-Riemannian model, the holonomy group is generated by non-integrability, and control protocols (e.g., isoparallel-mass-transport, IMT) can be designed to guarantee $g$6 ('no mixing'): tracer particles return to their original positions after traversing closed curves. This structure is fully developed in (Abdelgalil et al., 2024).

5. Interpolation Inequalities, Distortion Coefficients, and Metric Geometry

On ideal sub-Riemannian manifolds, distortion coefficients $g$7, defined via the Jacobian of the sub-Riemannian exponential map (using Jacobi fields), underpin sharp Brunn–Minkowski, Borell–Brascamp–Lieb, and interpolation inequalities. For $g$8,

$g$9

with $\Delta$0, and $\Delta$1 the unique normal geodesic from $\Delta$2 to $\Delta$3 at time $\Delta$4. The geodesic ‘dimension’ $\Delta$5 (generally $\Delta$6) quantifies the mass-spreading rate, deviating from the Riemannian case due to horizontal-vertical splitting and non-integrability (Barilari et al., 2017). Explicit exponent calculations are available for the Heisenberg group, generalized H-type Carnot groups, and the Grushin plane; for instance, the Brunn–Minkowski formula on $\Delta$7 involves exponent $\Delta$8 rather than $\Delta$9.

6. Sub-Riemannian Transport in Applied and Stochastic Settings

Sub-Riemannian transport structures naturally occur in complex systems with nonholonomic constraints, such as:

• Ocean neutral transport: Here, water parcels move along (locally defined) neutral planes—contact distributions $\gamma: [0,1] \to M$0, with $\gamma: [0,1] \to M$1 built from climatological gradients. The horizontal (neutral) constraint and non-integrability (helicity) yield global accessibility (by the Chow–Rashevskii theorem) and highly anisotropic geodesics. Stochastic analogues involve hypoelliptic Brownian diffusions tangent to $\gamma: [0,1] \to M$2; transition densities reflect the sub-Riemannian distance, and mixing times for dianeutral dispersion are estimated at $\gamma: [0,1] \to M$3centuries (Chatelain et al., 3 Nov 2025).
• Congested transport: For the Heisenberg group, the sub-Riemannian constraint restricts admissible paths to horizontal curves in the congested optimal transport problem, leading to modified Eulerian and Lagrangian formulations of traffic intensity and equilibrium (Wardrop) states (Circelli et al., 2023).
• Transport on path spaces and inequalities: Talagrand-type transport inequalities can be generalized to horizontal Brownian motion on step-2 Carnot groups, but projection arguments used in the Euclidean setting for path-space measures fail due to the non-commutativity and vertical blows-up. Riemannian approximations recover some inequalities for $\gamma: [0,1] \to M$4, but not at $\gamma: [0,1] \to M$5 (Friz et al., 6 Feb 2026).

7. Regularity, Cut Locus, and Advanced Metric Properties

The regularity theory for sub-Riemannian transport is complex: the squared distance function $\gamma: [0,1] \to M$6 is semiconvex away from the cut locus, which is larger and subtler than in the Riemannian case. The cut locus is characterized as the set where semiconvexity fails. The analysis of Jacobi fields and sub-Riemannian index forms provides comparison theorems (e.g., for Sasakian manifolds) and controls on the Hessian of the distance. In applied settings, these results enable comparison principles and coupling estimates for hypoelliptic diffusions and underpin entropy dissipation and $\gamma: [0,1] \to M$7-calculus for kinetic Fokker-Planck equations on sub-Riemannian density manifolds (Baudoin et al., 2022, Feng et al., 2019).

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References:

• (Abdelgalil et al., 2024) Sub-Riemannian Geometry, Mixing, and the Holonomy of Optimal Mass Transport
• (Badreddine, 2017) Mass Transportation on sub-Riemannian structures of rank two in dimension four
• (Circelli et al., 2023) Transport densities and congested optimal transport problem in the Heisenberg group
• (Friz et al., 6 Feb 2026) Talagrand-type transport inequalities for path spaces over Carnot groups
• (Citti et al., 28 Jul 2025) Benamou-Brenier and Kantorovich are equivalent on sub-Riemannian manifolds with no abnormal geodesics
• (Feng et al., 2019) Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold
• (Baudoin et al., 2022) Variations of the sub-Riemannian distance on Sasakian manifolds with applications to coupling
• (Barilari et al., 2017) Sub-Riemannian interpolation inequalities
• (Chatelain et al., 3 Nov 2025) Ocean neutral transport: sub-Riemannian geometry and hypoelliptic diffusion