Sub-Riemannian Manifolds

Roughly speaking, a sub-Riemannian manifold is a manifold that has measuring restrictions. We are allowed to measure the magnitude of vectors only for a distinguished subset of vectors, called horizontal vectors. In other words, we cannot observe the manifold in all directions, fact that allows for “missing” or “forbidden” directions. The existence of these directions is intimately related to the noncommutativity properties built on the manifold. More precisely, we have the following definition.

Definition 2.1.1.A sub-Riemannian manifold is a real manifold M of dimension n together with a nonintegrable distribution D of rank k (k < n) endowed with a sub-Riemannian metric g, i.e., an application gp: Dp × Dp → ℝ, for all p ∈ M, which is a positive de?nite, nondegenerate, inner product.

We note that the rank k cannot be equal to the dimension n because in this case the distribution D will be tangent to the manifold M and then Dx = TxM; i.e., the distribution D becomes integrable because M is an integral manifold.

The nonintegrability condition of the distribution D is essential. If the distribution were integrable, then, at least locally, there would be a submanifold S of M tangent to D, in which case (S, g) becomes a Riemannian manifold and the theory D falls in the case of a very well known class of manifolds.

One may denote the sub-Riemannian manifold by the triplet (M, D, g).

A few important discoveries in the field of thermodynamics in the 1800s made the first steps toward sub-Riemannian geometry. Carnot discovered the principle of an engine in 1824 involving two isotherms and two adiabatic processes, Jule studied adiabatic processes, and Clausius formulated the existence of the entropy in the second law of thermodynamics in 1854. In 1909 Carathéodory made the point regarding the relationship between the connectivity of two states by adiabatic processes and nonintegrability of a distribution, which is defined by the one-form of work. Chow proved the general global connectivity in 1934, and the same hypothesis was used by Hörmander in 1967 to prove the hypoellipticity of a sum of the squares of vector fields operators. However, the study of the invariants of a horizontal distribution, known as nonholonomic geometry, was initiated by the Romanian mathematician George Vranceanu in 1936.

The position of a ship on a sea is determined by three parameters: two coordinates x and y for the location and an angle to describe the orientation. Therefore, the position of a ship can be described by a point in a manifold. One can ask what is the shortest distance one should navigate to get from one position to another; this defines a Carnot–Carathéodory metric on the manifold ℝ2 × S1. In a similar way, a Carnot–Carathéodory metric can be defined on a general sub-Riemannian manifold. The study of sub-Riemannian geodesics is useful in determining the Carnot–Carathéodory distance between two points.

Definition and Examples

A Grushin manifold is roughly speaking a Riemannian manifold endowed with a singular Riemannian metric. In this case the horizontal distribution does not play an important role, since the distribution coincides with the tangent bundle on the set of regular points. The horizontal curves do not make much sense either in this case. A Grushin manifold might be considered as a sub-Riemannian manifold where the distribution has the rank equal to the dimension of the space. We study this type of manifold here since it behaves similarly with some of the examples studied in the previous chapters. They are also closely related with a certain type of subelliptic operators, called Grushin operators.

Definition 11.1.1.Let M be a manifold of dimension n and let X1,…, Xn be n vectors on M. Let S ={p ∈ M; span{X1,…, Xn} = Tp M}. A point p ∈ S is called singular, while a point p ∉ S is called regular. Consider the Riemannianc metric g defined on the set of regular points M\S such that g(Xi, Xj) = δij. Then (M, Xi, g) is called a Grushin manifold.

Recall that the step at a point p ∈ M is equal to 1 plus the number of Lie brackets of vector fields Xi needed to span the tangent space TpM.