Optimal transport, Euler equations, Mather and DiPerna-Lions theories

1.3. SUB-RIEMANNIAN MANIFOLDS 13
Theorem 1.2.3 If Ric ≥ 0 and A satisfies (1.2.1), then U is displacement convex.
In the above case, we have considered functional defined only on probability measures which
are absolutely continuous with respect to vol. The big advantage of this fact is that, in this
case, the curve µt we defined above is the unique Wasserstein geodesic between µ and ν. On
the other hand, if we do not make any absolute continuity assuptions, the Wasserstein geodesic
(which always exists) is in general not unique. One can therefore introduce two different notions
of dispacement convexity, a strong and a weak one: the strong notion consists in asking that a
functional defined in the space of probability measure on M is convex among all Wasserstein
geodesics connecting two measures µ and ν; the weak one is that, for all µ and ν, there is some
Wasserstein geodesic connecting them along which the functional is convex. The importance
of introducing a weaker notion comes from the fact that it is more stable under passage to the
limit, and so it is particularly suitable when one wants to recast lower bounds on the Ricci
curvature tensor in terms of displacement convexity properties of certain nonlinear functionals
[59, 73, 74]. However, as shown in collaboration with Cédric Villani, on Riemannian manifolds
these notions are equivalent [24].
1.3 Sub-Riemannian manifolds
A sub-Riemannian manifold is given by a triple (M, ∆, g) where M denotes a smooth complete
connected manifold of dimension n, ∆ is a smooth nonholonomic distribution of rank m < n on
M, and g is a Riemannian metric on M. We recall that a smooth distribution of rank m on M
is a rank m subbundle of T M. This means that, for every x ∈ M, there exist a neighborhood
Vx of x in M, and a m-tuple (f x 1 , . . . , f x m) of smooth vector fields on Vx, linearly independent on
Vx, such that
∆(z) = Span {f x 1 (z), . . . , f x m(z)} ∀z ∈ Vx.
One says that the m-tuple of vector fields (f x 1 , . . . , f x m) represents locally the distribution ∆.
We assume that the distribution ∆ is nonholonomic, i.e. for every x ∈ M there is a m-tuple
(f x 1 , . . . , f x m) of smooth vector fields on Vx which represents locally the distribution and such
that
Lie {f x 1 , . . . , f x m} (z) = TzM ∀z ∈ Vx,
that is, such that the Lie algebra 2 spanned by f x 1 , . . . , f x m, is equal to the whole tangent space
TzM at every point z ∈ Vx. This Lie algebra property is often called Hörmander’s condition.
A curve γ : [0, 1] → M is called a horizontal path with respect to ∆ if it belongs to
W 1,2 ([0, 1], M) and satisfies
˙γ(t) ∈ ∆(γ(t)) for a.e. t ∈ [0, 1].
2 We recall that, for any family F of smooth vector fields on M, the Lie algebra of vector fields generated by
F, denoted by Lie(F), is the smallest vector space S satisfying
where [X, Y ] is the Lie bracket of X and Y .
[X, Y ] ⊂ S ∀X ∈ F, ∀Y ∈ S,