Sub-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, are best understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, is one of the simplest examples of a sub-Riemannian manifold.

Definitions

By a distribution on $M$ we mean a subbundle of the tangent bundle of $M$ . Given a distribution $H(M)\subset T(M)$ a vector field in $H(M)\subset T(M)$ is called horizontal. A curve $\gamma$ on $M$ is called horizontal if $\dot\gamma(t)\in H_{\gamma(t)}(M)$ for any $t$ .

A distribution on $H(M)$ is called completely non-integrable if for any $x\in M$ we have that any tangent vector can be presented as a linear combination of vectors of the following types $A(x),\$ ^*(x),\ ^*]](x),...\in T_x(M) where all vector fields $A,B,C,D, ...$ are horizontal.

A sub-Riemannian manifold is a triple $(M, H, g)$ , where $M,$ is a differentiable manifold, $H$ is a completely non-integrable "horizontal" distribution and $g$ is a smooth section of positive-definite quadratic forms on $H$ .

Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot-Carathéodory, defined as

d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))}

, where infimum is taken along all horizontal curves

\to M

such that

\gamma(0)=x

\gamma(1)=y

Examples

A position of a car on the plane is determined by three parameters: two coordinates $x$ and $y$ for the location and an angle $\alpha$ which describes the orientation of the car. Therefore, the position of car can be described by a point in a manifold $\mathbb R^2\times S^1$ . 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 $\mathbb R^2\times S^1$ .

Closely related example of sub-Riemannian metric can be constructed on Heisenberg group: Take two elements in corresponding Lie algebra $\alpha,\beta$ , such that $\alpha,\beta,$ ^* span all algebra. Then horizontal distribution $H$ spanned by left shifts of $\alpha$ and $\beta$ is completely non-integrable. Then one has to choose any smooth positive quadratic form on $H$ .

Properties

For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the cometric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold. The existence of geodesics of the corresponding Hamilton-Jacobi equations for the sub-Riemannian Hamiltonian are given by the Chow-Rashevskii theorem.

References

Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9.

Metric geometry | Riemannian geometry

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Definitions

Examples

Properties

References