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 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 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 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
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