The last post of Tao from his series of posts on the Hilbert’s fifth problem contains interesting results which can be used for understanding the differences between Gleason distances and sub-riemannian distances or, more general, norms on groups with dilations.
There is also another post of mine on the Gleason metric (distance) and the CC (or sub-riemannian) distance, where I explain why the commutator estimate (definition 3, relation (2) from the last post of Tao) forces “commutativity”, in the sense that a sub-riemannian left invariant distance on a Lie group which has the commutator estimate must be a riemannian distance.
What about the escape property (Definition 3, relation (1) from the post of Tao)?
From his Proposition 10 we see that the escape property implies the commutator estimate, therefore a sub-riemannian left invariant distance with the escape property must be riemannian.
An explanation of this phenomenon can be deduced by using the notion of “coherent projection”, section 9 of the paper
in the very particular case of sub-riemannian Lie groups (or for that matter normed groups with dilations).
Suppose we have a normed group with dilations which has another left invariant dilation structure on it (in the paper this is denoted by a “ bar”, here I shall use the notation for this supplementary dilation structure).
There is one such a dilation structure available for any Lie group (notice that I am not trying to give a proof of the H5 problem), namely for any (but not too big)
(maybe interesting: which famous lemma is equivalent with the fact that is a group with dilations?)
Take to be a dilation structure coming from a left-invariant distribution on the group . Then commutes with and moreover
where is a projection: for any .
It is straightforward to check that (the left-translation of) (over the whole group) is a coherent projection, more precisely it is the projection on the distribution!
Exercise: denote by and use (*) to prove that the escape property of Tao implies that is (locally) injective. This implies in turn that , therefore the distribution is the tangent bundle, therefore the distance is riemannian!
UPDATE: See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for understanding in detail the role of the escape property in the proof of the Hilbert 5th problem.