Sub-Riemannian manifold

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

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

A distribution H ( M ) {\displaystyle H(M)} {\displaystyle H(M)} is called completely non-integrable or bracket generating if for any x ∈ M {\displaystyle x\in 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)} {\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 } {\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)} {\displaystyle (M,H,g)}, where M {\displaystyle M} {\displaystyle M} is a differentiable manifold, H {\displaystyle H} {\displaystyle H} is a completely non-integrable "horizontal" distribution and g {\displaystyle g} {\displaystyle g} is a smooth section of positive-definite quadratic forms on H {\displaystyle 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,} {\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} {\displaystyle \gamma :[0,1]\to M} such that γ ( 0 ) = x {\displaystyle \gamma (0)=x} {\displaystyle \gamma (0)=x}, γ ( 1 ) = y {\displaystyle \gamma (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)} {\displaystyle 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.

Examples

A position of a car on the plane is determined by three parameters: two coordinates x {\displaystyle x} {\displaystyle x} and y {\displaystyle y} {\displaystyle y} for the location and an angle α {\displaystyle \alpha } {\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}.} {\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}.} {\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 } {\displaystyle \alpha } and β {\displaystyle \beta } {\displaystyle \beta } in the corresponding Lie algebra such that

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

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

Properties

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

References