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Sub-Riemannian manifold

Type of generalization of a Riemannian manifold

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated September 13, 2026

In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces.

Sub-Riemannian manifolds (and so, a fortiori, Riemannian manifolds) carry a natural intrinsic metric called the metric of Carnot-Carathéodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold).

Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure.

01Definitions

By a distribution on M we mean a subbundle of the tangent bundle of M (see also distribution).

Given a distribution H(M)\subseteq T(M) a vector field in H(M) is called horizontal. A curve \gamma on M is called horizontal if {\dot {\gamma }}(t)\in H_{\gamma (t)}(M) for any t.

A distribution H(M) is called completely non-integrable or bracket generating if for any x\in M we have that any tangent vector can be presented as a linear combination of Lie brackets of horizontal fields, i.e. vectors of the form A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc \in T_{x}(M) where all vector fields A,B,C,D,\dots are horizontal. This requirement is also known as Hörmander's condition.

A sub-Riemannian manifold is a triple (M,H,g), where M is a differentiable manifold, H is a completely non-integrable "horizontal" distribution and g is a smooth section of positive-definite quadratic forms on H.

Any (connected) sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot-Carathéodory, defined as

d(x,y)=\inf \int _{0}^{1}{\sqrt {g({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt,

where infimum is taken along all horizontal curves \gamma :[0,1]\to M such that \gamma (0)=x, \gamma (1)=y. Horizontal curves can be taken either Lipschitz continuous, absolutely continuous or in the Sobolev space H^{1}([0,1],M) producing the same metric in all cases.

The fact that the distance between two points is always finite (i.e. any two points are connected by a horizontal curve) is a consequence of Hörmander's condition known as Chow-Rashevskii theorem.

02Examples

A position of a car on the plane is determined by three parameters: two coordinates x and y for the location and an angle \alpha which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold

\mathbb {R} ^{2}\times S^{1}.

One can ask, what is the minimal distance one should drive to get from one position to another? This defines a Carnot-Carathéodory metric on the manifold

\mathbb {R} ^{2}\times S^{1}.

A closely related example of a sub-Riemannian metric can be constructed on a Heisenberg group: Take two elements \alpha and \beta in the corresponding Lie algebra such that

\{\alpha ,\beta ,[\alpha ,\beta ]\}

spans the entire algebra. The distribution H spanned by left shifts of \alpha and \beta is completely non-integrable. Then choosing any smooth positive quadratic form on H gives a sub-Riemannian metric on the group.

03Properties

For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold.

Solutions of the corresponding Hamilton-Jacobi equations for the sub-Riemannian Hamiltonian are called geodesics, and generalize Riemannian geodesics.

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Sources and credits

This article is adapted from the Wikipedia article “Sub-Riemannian manifold”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Carnot group

In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal.

Lie group

In mathematics, a Lie group is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction.

Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold M {\displaystyle M} is an assignment x ↦ Δ x ⊆ T x M {\displaystyle x\mapsto \Delta _{x}\subseteq T_{x}M} of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle T M {\displaystyle TM} .