Sub-Riemannian manifold

Type of generalization of a Riemannian manifold

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.

## Definitions

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

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

A distribution on  H ( M ) {\displaystyle H(M)} is called completely non-integrable or bracket generating if for any  x ∈ M {\displaystyle 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 ) , … ∈ T x ( M ) {\displaystyle 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 , … {\displaystyle 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 ) {\displaystyle (M,H,g)} , where  M {\displaystyle M} is a differentiable manifold,  H {\displaystyle H} is a completely non-integrable "horizontal" distribution and  g {\displaystyle g} is a smooth section of positive-definite quadratic forms on  H {\displaystyle H} .

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

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

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

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

## Examples

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

R 2 × S 1 . {\displaystyle \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

R 2 × S 1 . {\displaystyle \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  α {\displaystyle \alpha } and  β {\displaystyle \beta } in the corresponding Lie algebra such that

{ α , β , [ α , β ] } {\displaystyle \\{\alpha ,\beta ,[\alpha ,\beta ]\\}}

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

## Properties

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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.

## See also

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* Carnot group, a class of Lie groups that form sub-Riemannian manifolds.
* Distribution
* Hörmander's condition
* Optimal control

## References

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* Agrachev, Andrei; Barilari, Davide; Boscain, Ugo, eds. (2019), Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, Cambridge University Press, doi:10.1017/9781108677325, ISBN 9781108677325
* Bellaïche, André; Risler, Jean-Jacques, eds. (1996), Sub-Riemannian geometry, Progress in Mathematics, vol. 144, Birkhäuser Verlag, ISBN 978-3-7643-5476-3, MR 1421821
* Gromov, Mikhael (1996), "Carnot-Carathéodory spaces seen from within", in Bellaïche, André; Risler., Jean-Jacques (eds.), Sub-Riemannian geometry (PDF), Progr. Math., vol. 144, Basel, Boston, Berlin: Birkhäuser, pp. 79–323, ISBN 3-7643-5476-3, MR 1421823, archived from the original (PDF) on July 9, 201579-323&rft.pub=Birkhäuser&rft.date=1996&rft.isbn=3-7643-5476-3&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1421823#id-name=MR&rft.aulast=Gromov&rft.aufirst=Mikhael&rft_id=https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf&rfr_id=info:sid/en.wikipedia.org:Sub-Riemannian+manifold" class="Z3988">
* Le Donne, Enrico, Lecture notes on sub-Riemannian geometry (PDF)
* Montgomery, Richard (2002), A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, ISBN 0-8218-1391-9

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Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main results (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
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* Local
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* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Closed
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* (Almost) Complex
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* Fibered
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* G-structure
* Hadamard
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* Lie group
* Lie algebra
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* Parallelizable
* Poisson
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* Quaternionic
* Hypercomplex
* (Pseudo−, Sub−) Riemannian
* Rizza
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Vector-valued
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach manifold
* Diffeology
* Diffiety
* Fréchet manifold
* K-theory
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
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Riemannian geometry (Glossary)

Basic concepts|

* Curvature
* tensor
* Scalar
* Ricci
* Sectional
* Exponential map
* Geodesic
* Inner product
* Metric tensor
* Levi-Civita connection
* Covariant derivative
* Signature
* Raising and lowering indices/Musical isomorphism
* Parallel transport
* Riemannian manifold
* Pseudo-Riemannian manifold
* Riemannian volume form

Types of manifolds|

* Hermitian
* Hyperbolic
* Kähler
* Kenmotsu

Main results|

* Fundamental theorem of Riemannian geometry
* Gauss's lemma
* Gauss–Bonnet theorem
* Hopf–Rinow theorem
* Nash embedding theorem
* Ricci flow
* Schur's lemma

Generalizations|

* Finsler
* Hilbert
* Sub-Riemannian

Applications|

* General relativity
* Geometrization conjecture
* Poincaré conjecture
* Uniformization theorem

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