nLab sub-Riemannian geometry

Idea

Sub-Riemannian geoemtry is a generalization of Riemannian geometry where the metric tensor is not necessarily defined on the whole tangent bundle but on a smooth sub-bundle (a distribution of subspaces), then called the subbundle of horizontal vector fields.

Typically authors require further conditions on the horizontal subbundle, notably the bracket-generating condition (also called the Hörmander condition or Chow-Rashevskii condition) which states that the operation of forming iterated Lie brackets of germs of horizontal vector fields yields all germs of all vector fields.

Related entries

• Heisenberg manifold

References

Monograph:

• Ovidiu Calin, Der-Chen Chang: Sub-Riemannian Geometry – General Theory and Examples, Cambridge University Press (2009) 175–230 [doi:10.1017/CBO9781139195966]

Lecture notes:

• Enrico Le Donne: Lecture notes on sub-Riemannian geometry — from the Lie group viewpoint (2021) [pdf]

See also:

• Wikipedia: Sub-Riemannian manifold

• Wikipedia: Bracket-generating distribution