Math 2.0

Differential geometry Definition

Sub-Riemannian manifold

Builds on Smooth manifoldTangent bundleLie bracket of vector fieldsInner product

Definition

Let MM be a connected smooth manifold with tangent bundle TMTM. A sub-Riemannian manifold is a triple (M,H,g)(M, H, g) such that:

  1. H⊆TMH \subseteq TM is a smooth subbundle, the horizontal distribution;
  2. HH is bracket-generating: at each p∈Mp \in M, the vector fields tangent to HH and their iterated Lie brackets span TpMT_pM;
  3. gg assigns to each p∈Mp \in M an inner product gpg_p on the fibre HpH_p, depending smoothly on pp.

A horizontal curve is an absolutely continuous curve γ\gamma with γ′(t)∈Hγ(t)\gamma'(t) \in H_{\gamma(t)} for almost every tt. The sub-Riemannian distance d(p,q)d(p, q) is the infimum of ∫g(γ′,γ′) dt\int \sqrt{g(\gamma', \gamma')} \, dt over the horizontal curves from pp to qq.

Remarks

Example: the Heisenberg group is ℝ3\R^3 with HH spanned by X=∂x−y2∂zX = \partial_x - \tfrac{y}{2} \partial_z and Y=∂y+x2∂zY = \partial_y + \tfrac{x}{2} \partial_z, with X,YX, Y orthonormal. Here [X,Y]=∂z[X, Y] = \partial_z, so X,Y,[X,Y]X, Y, [X, Y] span Tpℝ3T_p\R^3 at each point, and clause 2 holds.

Non-example: on ℝ3\R^3, let HH be spanned by ∂x\partial_x and ∂y\partial_y. All brackets of fields tangent to HH stay in HH, so clause 2 fails; horizontal curves stay in the planes z=constz = \text{const}, and points with different zz are at infinite distance.

When H=TMH = TM, the definition gives a Riemannian manifold. Some authors do not require clause 2 in the definition and add it as a hypothesis when needed.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Inner product, concept, no evidence levelInner productconceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTangent bundle, concept, no evidence levelTangent bundleconceptSub-Riemannian manifold, definition, no evidence levelSub-Riemannian manifo…definition
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  1. Inner product concept
  2. Lie bracket of vector fields concept
  3. Smooth manifold concept
  4. Tangent bundle concept
  5. Sub-Riemannian manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Sub-Riemannian_manifold
About this page
TagKK25XTQG41
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Kinddefinition
WritingW0 stub
MSC53C17
DefinesSub-Riemannian manifold
UsesSmooth manifold, Tangent bundle, Lie bracket of vector fields, Inner product
Changed2 November 2026

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