Cometrics on Sub-Riemannian manifolds

[Kreizhn]

Hi all,

This may seem like a simple problem, but I just want to clarify something.

The issue is the relationship between sub-Riemannian manifolds and cometrics. In particular, say we have a manifold M and a cometric on the cotangent bundle T*M. Firstly, it is my understanding that somehow a cometric "induces" a sub-Riemannian manifold; I think this is done by defining the Hamiltonian
$$H(p,q) = \frac12 \langle p,p\rangle, (p,q) \in T_q^*M$$
and using the fact that all sub-Riemannian manifolds are completely characterized by their Hamiltonian. Is this correct?

Secondly, can one go in the other direction. Namely, if one has a sub-Riemannian manifold, is there a naturally defined cometric? If so, can it be defined on the entire cotangent bundle or just some sort of dual to the horizontal distribution?

Naturally above, I'm considering the case when the distribution is not the entire tangent bundle, and when the cometric is degenerate. Otherwise it would seem obvious that the musical isomorphisms would give the necessary relations.

 

[jvicens]

Hello,

You are correct in your understanding that a cometric on the cotangent bundle T*M induces a sub-Riemannian manifold. This is because a cometric defines a metric on the horizontal distribution, which is a subbundle of the tangent bundle. This metric can then be used to define a sub-Riemannian structure on the manifold.

As for going in the other direction, it is possible to define a cometric from a sub-Riemannian manifold, but it may not be unique. In general, there is not a unique cometric that corresponds to a given sub-Riemannian structure. The cometric can be defined on the entire cotangent bundle, but it may not be degenerate. If the sub-Riemannian structure is not completely nonholonomic, then the cometric may be degenerate.

I hope this clarifies your understanding of the relationship between sub-Riemannian manifolds and cometrics. Let me know if you have any further questions.