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Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Sub-Riemannian curvature: a variational approach
Davide Barilari (Université Paris Diderot - Paris 7)
IHP Workshop “Geometric Analysis on Sub-Riemannian manifolds”
October 2, 2014
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 1 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
References
This is a joint work with
Andrei Agrachev (SISSA, Trieste)
Luca Rizzi (CMAP, Ecole Polytechnique)
→ References:
- A. Agrachev, D.B., L.Rizzi, The curvature: a variational approach.ArXiv preprint 2013, 114 p.
- D.B., L. Rizzi, Comparison theorems for conjugate points in sub-Riemanniangeometry, Arxiv preprint 2014.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 1 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 1 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 1 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Introduction
In Riemannian geometry the curvature tensor plays a major role in many aspects:
Ricci bounds
comparison theorems (volume, distance, Laplacian, Hessian etc.)
asymptotic and estimates of the heat kernel
Obstruction 1. In sub-Riemannian geometry no canonical affine connection
Recently, many efforts to generalize the notion of Ricci/curvature bounds
via optimal transport techniques
Lott, Villani - SturmAmbrosio, Gigli, Savaré (and coauthors)
via gen. curvature dimension inequalities / heat equation techniques
Garofalo, Baudoin (and coauthors)Thalmaier, Grong
via measure contraction properties / other geometric techniques
Julliet - Rifford - VassilevAgrachev, Lee - Lee, Li, Zelenko
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 2 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Introduction II
In this talk I will be interested in looking for a definition/generalization of thenotion of (sectional) curvature in sub-Riemannian geometry.
This curvature is based on two main tools
distance (→ or more generally a cost)
geodesics (→ or more generally minimizers for the cost)
From these two ingredients we define a “directional curvature”.
Naive idea:
→ The curvature can be found in the asymptotic of the distance along geodesics
Obstruction 2. Non-smoothness of the distance in sub-Riemannian geometry
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 3 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Riemannian sectional curvature
Let (M , d) be a Riemannian manifold and x ∈ M .
Let γv , γw two geodesics with unit tangent vector v ,w ∈ TxM .
x
γv (t) γw(s)
C (t, s) := 12d
2(γv(t), γw (s))
is smooth in (0, 0)bc
bc bc
We expect to recover the scalar product and the curvature from the asymptoticexpansion of C (t, s).
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 4 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Asymptotic expansion
C (t, s) ≃1
2(t2 + s2)− 〈v ,w〉 ts
︸ ︷︷ ︸
Euclidean
+1
4∂ttssC (0, 0)t2s2 + . . .
Theorem (Loeper, Villani)
−3
2∂ttssC (0, 0) =
⟨R∇(v ,w)v ,w
⟩(= Sec(v ,w))
Remarks
→ In SR geometry the geodesic is not uniquely associated with its tangentvector.
→ The function C (t, s) is not smooth at (0,0).
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 5 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Lemma: differential of d2(x0, ·)
Let x0 ∈ M and define
fx0(·) =1
2d2(x0, ·)
x0
γ : [0,T ] → M b
x
γ′v(T )
v
fx0 smooth at x ⇐⇒ ∃ ! minimal non conjugate geodesic γ joining x0 and x .
∇fx0(x) = Tγ′v(T )
Remark. If we define ct(x) := − 1
2td2(x , γv (t)) then ∇ct(x0) = v (indep. on t).
∇x0
(
−1
2td2(·, γv (t))
)
= −1
t∇fγv (t)(x0) = −
1
t(−tv) = v
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 6 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 6 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Affine optimal control problems
[Dynamic] Let us consider a smooth affine control system on a manifold M
x = X0(x) +
k∑
i=1
uiXi (x), x ∈ M , u ∈ Rk .
we call Dx = spanxX1, . . . ,Xk the distribution.
u ∈ L∞([0,T ],Rk) → γx0,u(·) the solution with x(0) = x0 (admissible curve)
[Cost] Given a smooth function L : M × Rk → R we define the cost at time T as
the functional
JT (u) :=
∫ T
0
L(γu(t), u(t))dt,
Definition
For two given points x , y ∈ M and T > 0, we define the value function
ST (x , y) = inf JT (u) | u admissible, γu(0) = x , γu(T ) = y ,
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 7 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Assumptions
(A1) The weak bracket generating condition is satisfied
Liex(adX0)
iXj , i ∈ N, j = 1, . . . , k= TxM ,
[the vector field X0 is not included in the generators of the Lie algebra]
(A2) The Lagrangian L : M × Rk → R is of Tonelli-type, i.e.
(A2.a) L has strictly positive Hessian in u, for all x ∈ M.(A2.b) L has superlinear growth, i.e. L(x , u)/|u| → +∞ when |u| → +∞.
(A1) does not ask that the linear part is bracket-generating.
→ when X0 = 0 it is the classical Hormander condition.
(A2) ensures that the Hamiltonian is well defined and smooth
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 8 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Geometric framework
This setting includes control problems associated with many different geometricstructures such as
Riemannian structuressub-Riemannian structuressmooth Finsler structures (or smooth sub-Finsler).
The (sub-)Riemannian case corresponds to the case when
the system is driftless (X0 ≡ 0) and bracket generating (A1)
k < n (k = n corresponds to Riemannian)
the cost is quadratic L(x , u) = 1
2|u|2 (A2)
→ The cost is induced by a scalar product such that X1, . . . ,Xk are orthonormal.
ST (x , y) =1
2Td2(x , y)
In particular
ct(x) = −1
2td2(x , γ(t)) = −St(x , γ(t))
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 9 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Main idea: microlocal approach
Fix x0 ∈ M and a minimizing trajectory γ = γx0,u(t).
ct : M → R, t > 0
ct(x) = −St(x , γ(t))
x0
γ(t)
ct : x 7→ −St(x , γ(t))
b
x
The curvature appears in the asymptotic expansion of the second derivative of thecost along a minimizing trajectory.
We need some assumptions on γ to guarantee
the smoothness of ct (→ strongly normal)
the existence of the asymptotic. (→ ample)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 10 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
End point map
Definition
Fix a point x0 ∈ M and T > 0. The end-point map at time T is the map
Ex0,T : U → M , u 7→ γx0,u(T ),
For every x0, x1 ∈ M one has
ST (x0, x1) = infE
−1x0,T
(x1)JT = infJT (u) |Ex0,T (u) = x1
In general Ex0,T is smooth but
DuEx0,T is not surjective
the set E−1
x0,T(x1) is not a smooth manifold.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 11 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Lagrange multipliers rule
A necessary condition for constrained critical point of infE−1x0,T
(x1)JT .
Theorem
Assume u ∈ U is a constr. crit. point, with x = Ex0,T (u). Then (at least) one ofthe two following statements hold true
(i) ∃λT ∈ T ∗x M s.t. λT · DuEx0,T = DuJT , λT · DuEx0,T = DuJT ,
(ii) ∃λT ∈ T ∗x M s.t. λT · DuEx0,T = 0 (here λT 6= 0).
→ A control u (reps. the associated trajectory γu) is called
normal in case (i),abnormal in case (ii).
A priori a single control can be associated with two covectors such that both (i)and (ii) are satisfied.
In what follows we will call geodesic any normal extremal.
A geodesic γ : [0,T ] → M is called strongly normal if for all s ∈ [0,T ] therestriction γ|[0,s] is not abnormal.Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 12 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Hamiltonian and PMP for normal
H(λ) = maxu∈Rk
(〈λ,Xu(x)〉 − L(x , u)) , λ ∈ T ∗M , x = π(λ).
where Xu(x) = X0(x) +∑
uiXi (x)
Theorem (PMP, normal case)
Let (u(t), γu(t)) be a normal geodesic. Then there exists a Lipschitz curveλ(t) ∈ T ∗
γu(t)M such that
λ(t) =−→H (λ(t), u(t)),
Any normal geodesic starting at x0 is characterized by its initial covectorλ0 ∈ T ∗
x0M and we write
γ(t) = Expx0(t, λ0)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 13 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Distance from a minimizer
Fix x0 ∈ M and a strongly normal geodesic γ(t) = Expx0(t, λ0) and definethe geodesic cost
ct : M → R, t > 0
ct(x) = −St(x , γ(t))
x0
γ(t)
ct : x 7→ −St(x , γ(t))
b
x
Theorem
There exist ε > 0 and an open set U ⊂ (0, ε)×M such that (t, x0) ∈ U
(i) The geodesic cost function (t, x) 7→ ct(x) is smooth on U .
(ii) For every (t, x) ∈ U there exists a unique minimizer connecting x and γ(t)
Moreover dx0ct = λ0 for any t ∈ (0, ε).
Then dx0 ct = 0 → d2x0ct : Tx0M → R is a well defined family of quadratic forms.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 14 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Riemannian case
Fix any Riemannian geodesic γ ⇒ well defined quadratic form d2x0ct : Tx0M → R
Define the family of operators Qγ(t) : Tx0M → Tx0M by
d2
x0ct(v) = 〈Qγ(t)v , v〉, v ∈ Tx0M , t ∈ (0, ǫ)
smooth family of symmetric operators for t ∈ (0, ǫ)
singular at t = 0
Regularity theorem (Riemannian case)
The family of symmetric operators has a second order pole at t = 0 and
Qγ(t) =1
t2I+
1
3R∇(γ, ·)γ + O(t)
→ R∇(γ, ·)γ can be though as “directional curvature operator” in the direction γ
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 15 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Interpretation for ct(x) in Riemannian geometry
Recall that, for a Riemannian structure
ct(x) := −St(x , γ(t)) = −1
2td2(x , γ(t))
Lemma
For a Riemannian structure, the derivative of the geodesic cost function is
ct(x) =1
2‖W t
x0,γ(t)−W t
x,γ(t)‖2 −
1
2
where W tx,y ∈ TyM is the tangent vector at time t of the unique minimizer
connecting x with y in time t
It encodes the curvature without using parallel transport.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 16 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 17 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 17 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Geodesic growth vector
Let γ be smooth admissible curve. Let T ∈ Vec(M) an admissible extension of γ
Geodesic flag
F iγ(t) = span[T , . . . , [T
︸ ︷︷ ︸
j≤i−1 times
,X ]]|γ(t) | X ∈ Γ(D), j = 0, . . . , i − 1
These subspaces define a natural flag
F1
γ(t) ⊂ F2
γ(t) ⊂ . . . ⊂ Tγ(t)M
F1γ(t) is the distribution Dγ(t).
Does not depend on the choice of TDepends on the germ of γ and of the distribution at γ(t) (→ microlocal)
Geodesic growth vector
Gγ(t) = k1(t), k2(t), . . ., ki (t) = dimF iγ(t)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 18 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Ample geodesic
Definition (Ample geodesic)
A geodesic γ is ample at t if ∃m = m(t) > 0 s.t.
F1
γ(t) ⊂ . . . ⊂ Fmγ (t) = Tγ(t)M
An ample geodesic is equiregular if its growth vector does not depend on t.
is a “microlocal Hörmander condition”.
ample at t = 0 ⇒ strongly normal
Theorem (The generic normal SR geodesic is ample at t = 0)
Let M be a sub-Riemannian manifold. For every x0 ∈ M there exists a non-emptyopen Zariski Ax0 ⊂ T ∗
x0M such that, for all λ ∈ Ax0 , the geodesic γ(t) = exp(tλ)
is ample at t = 0.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 19 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Hamiltonian scalar product on Dx
In the Riemannian case we used the scalar product to transform quadraticforms in operators.
In the sub-Riemannian case we have a scalar product on 〈·, ·〉 on Dx
→ restriction of our quadratic forms d2x0ct to Dx0
This is a scalar product induced by the Hamiltonian H . In the general case
L is Tonelli (i.e. d2uL non degenerate)
the second derivative of H |T∗
x M at λ defines a scalar product on Dx
- d2
λHx : T∗
x M → TxM self-adjoint linear map
- Im(d2
λHx ) = Dx for every λ ∈ T∗
x M.
〈v1|v2〉λ :=⟨
ξ1, d2
λHx (ξ2)
⟩
, d2
λHx (ξi ) = vi .
Scalar product 〈·, ·〉λ on Dx , depending on λ.
if H is (quadratic) sub-Riemannian, then it is the SR scalar product (not depon λ!).
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 20 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Main result
Consider the restriction d2x0ct∣∣Dx0
: Dx0 → R to the distribution.
→ the scalar product 〈·, ·〉λ on Dx0 let us to define a family of symmetricoperators
Qλ(t) : Dx0 → Dx0 , d2
x0ct(v) = 〈Qλ(t)v , v〉λ , v ∈ Dx0
Theorem
Assume the geodesic is ample. Then Qλ(t) has a second order pole at t = 0 and
Qλ(t) ≃1
t2Iλ +
1
3Rλ + O(t), for t → 0
where
Iλ ≥ I > 0
Rλ is the curvature operator. We def Ric(λ) := trace(Rλ).
→ Iλ and Rλ are operators defined on Dx0 (symmetric wrt 〈·, ·〉λ).Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 21 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Riemannian and Finsler
Riemannian
any geodesic is ample with trivial growth vector Gγ = n
Iλ = I
Rλ = R∇(γ, ·)γ
Finsler
any geodesic is ample with trivial growth vector Gγ = n
Iλ = I
Rλ = RFγ - Flag curvature operator of Finsler manifolds
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 22 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
The Heisenberg case
The Heisenberg group R3 = (x , y , z) with standard left-invariant structure
X = ∂x −y
2∂z , Y = ∂y +
x
2∂z
Every (non trivial) geodesic is
ample and equiregular
with geodesic growth vector G = (2, 3).
If one fix two geodesics γλ(t), γη(s) corresponding to two covectors λ, η
C (t, s) := 1
2d2(γλ(t), γη(s)) is not C 2 at zero!
Still we can determine the main expansion
Qλ(t) ≃1
t2Iλ +
1
3Rλ + O(t), for t → 0
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 23 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
The Heisenberg case
We compute it on the orthonormal basis v := γ(0) and v⊥ := γ(0)⊥ for Dx0 .
The matrices representing Iλ and Rλ in the basis v , v⊥ of Dx0 are
Iλ =
(1 00 4
)
, Rλ =2
5
(0 00 h2
z
)
, (1)
where λ has coordinates (hx , hy , hz) dual to o.n. basis X ,Y + Reeb Z
anisotropy of the different directions on Dx0
curvature is always zero in the direction of γ
curvature of lines hz = 0 is zero.
curvature of lift of circles hz 6= 0 is not bounded (nor constant)
→ trace(Iλ) = 5 ↔ is related to MCP(0,5).
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 24 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Contact 3D case
Every (non trivial) geodesic is ample and equiregular with geodesic growth vectorG = (2, 3).
The matrices representing Iλ and Rλ in the basis γ, γ⊥ of Dx0 are
Iλ =
(1 00 4
)
, Rλ =2
5
(0 00 rλ
)
, (2)
whererλ = h2
z + κ(h2
x + h2
y) + 3χ(h2
x − h2
y )
and λ has coordinates (hx , hy , hz) dual to o.n. basis X ,Y + Reeb Z .
χ and κ are two functions on M (χ = κ = 0 in Heisenberg)
In high dim contact case it is computed/computable (→ in preparation)
Using generalization of Jacobi fields
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 25 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Young Diagram for equiregular geodesics
Assume now that the geodesic is also equiregular: [← no dependence on t ]
Define di := ki − ki−1 = dimF iγ − dimF i−1
γ
→ By equiregularity one gets d1 ≥ d2 ≥ . . . ≥ dm,
d1
d2
dm−1
∑mi=1
di = n = dimM
dm
n1
n2
nk
Denote nj := length of the j-th row (j = 1, . . . , k)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 26 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
The operator Iλ
Being an operator on a Euclidean space Iλ : Dx0 → Dx0 has well definedeigenvalues and trace.
Theorem
Assume the geodesic is ample and equiregular. Then
spec Iλ = n21, . . . , n2
k
tr Iλ = n21+ . . .+ n2
k .
In particular we can rewrite
tr Iλ = d1 + 3d2 + 5d3 + . . .+ (2m− 1)dm
Notice that k1 = dimDx0 is the dimension of the distribution.
In the Riemannian we recover Iλ is the identity.
In the Heisenberg (and any 3D contact) trIλ = 5 for every geodesic.
looks like an “Hausdorff dimension” with odd coeff.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 27 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Why “a variational approach”?
Two main reasons:Tonelli Lagrangian → include also Finsler case, and not restrict to(sub)-Riemannian situationsDrift → To have more space to find constant curvature model
In our setting, even Heisenberg has no constant curvature(Rλ not constant with respect to λ)
Linear-Quadratic control system in Rn
x = Ax + Bu, x ∈ Rn, u ∈ R
k , JT (x(·)) =1
2
∫ T
0
|u|2 − 〈Qx , x〉 dt.
with A,B in normal form*, has constant curvature equal to Q.
(*) A, B are in Brunowsky normal form [i.e. systems of the form x (n) = u]
(**) controllability (Kalman) → any geodesic is ample and equiregular
dim Fi = rank(B,AB, . . . ,AiB).
(***) spec(Iλ) = Kronecker indices of Brunovsky normal form
In the paper with L. Rizzi we use these models of “constant curvature” to findDavide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 28 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 28 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
SR case and sub-Laplacian operator
A sub-Riemannian manifold M with o.n. frame Dx = spanxX1, . . . ,Xk
horizontal gradient ∇f =∑k
i=1Xi (f )Xi
Given a smooth measure µ we can introduce a sub-Laplacian operator
∆µ = divµ ∇ =
k∑
i=1
X 2
i + (divµXi )Xi
it is essentially self-adjoint on C∞0(M) w.r.t. µ
if the structure is Riemannian, this is the Laplace-Beltrami operator
Corollary (of the Main Theorem)
∆ct∣∣x0
≃tr Iλt2
+1
3Ric(λ) + O(t), for t → 0
→ does not depend on µ since ct has critical point at x0.→ it is just the trace of the main expansion ∆ct
∣∣x0
= trace(d2x0ct)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 29 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Application: sub-Laplacian of SR distance
We want a formula for the sub-Laplacian of the distance from a geodesic.
ft(·).=
1
2d2(·, γ(t)) = −tct(·), t ∈ (0, ε)
ft has no critical point at x0 → the volume should appear
Theorem
Assume dim Dx locally constant at x0 and γ(t) = expx0(tλ) equiregular.Then there exists a smooth n-form ω defined along γ, such that for any smoothvolume µ s.t. µγ(t) = eg(t)ωγ(t), we have
∆µft |x0 = trIλ − g(0)t −1
3Ric(λ)t2 + O(t3),
Theorem
Let γ be an equiregular geodesic with initial covector λ ∈ T ∗x0M . Then there
exists a smooth n-form ω defined along γ, such that for any smooth volume µ s.t.µγ(t) = eg(t)ωγ(t), we have
1Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 30 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Outline
1 Motivation
2 Affine control systems and geodesic cost
3 Geodesic growth vector, ample geodesics and curvature
4 Applications: Laplacian of the sub-Riemannian distance
5 Applications: SR geodesic dimension and MCP
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 30 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Sub-Riemannian homotheties
M is a complete, connected, orientable sub-Riemannian manifold
µ smooth volume form on M .
Denote Σx0 is the open and dense set where the function f = 1
2d2(x0, ·) is smooth.
The sub-Riemannian homothety is the map φt : Σx0 → M
φt(x) = γx0,x(t) = π e(t−1)~H(dx f).
Fix Ω ⊂ Σx0 be a bounded, measurable set, with 0 < µ(Ω) < +∞
let Ωx0,t.= φt(Ω).
The map t 7→ µ(Ωx0,t) is smooth
µ(Ωx0,t) → 0 for t → 0.
? At which order µ(Ωx0,t) ∼ t?
the order does not depend on the smooth measure µ!
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 31 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
x0
x
Ωφt(x)
b
b
b
Ωx0,t = φt(Ω)
Figure : Sub-Riemannian homothety of the set Ω with center x0.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 32 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Riemannian vs sub-Riemannian
Two basic examples
(R) For a Riemannian structure, it is well known that
µ(Ωx0,t) ∼ tdimM , for t → 0,
[here f (t) ∼ g(t) means f (t) = g(t)(C + o(1)) for t → 0 and C > 0]
(SR) In the Heisenberg group it follows from [Juillet, 2009] that
µ(Ωx0,t) ∼ t5, for t → 0,
5 is neither the topological (dim = 3) nor the metric dimension (dimH = 4).
→ The exponent is a different dimensional invariant, associated with behavior ofgeodesics based at x0.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 33 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Geodesic dimension
Definition
Let γ(t) = Exp(t, λ) be ample (at t = 0) with Gγ = k1, k2, . . . , km. Then wedefine
Nλ.=
m∑
i=1
(2i − 1)(ki − ki−1),
and Nλ.= +∞ if the geodesic is not ample.
→ If λ is associated with an equiregular geodesic γ then Nλ = trace Iλ.
The function λ 7→ Nλ is constant a.e. on T ∗x0M , assuming its minimum value.
Nx0
.= minNλ|λ ∈ T ∗
x0M < +∞.
We call Nx0 the geodesic dimension of M at x0.
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 34 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Main result
For every x0 ∈ M we have the inequality Nx0 ≥ dimM and the equality holdsif and only if the structure is Riemannian at x0.
If the distribution is equiregular at x0 we have Nx0 > dimH M from Mitchell’sformula.
Theorem
Let µ be a smooth volume. For any bounded, measurable set Ω ⊂ Σx0 , with0 < µ(Ω) < +∞ we have
µ(Ωx0,t) ∼ tNx0 , for t → 0. (3)
In general this number can depend on the point
If the structure is left invariant is it does not.
Nx0 = 5 in 3D contact and Nx0 = 2n+ 3 in (2n+ 1)-dim contact manifolds
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 35 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
An open question: MCP on Carnot Groups
We say that an n-dim manifol M satisfies MCP(0,D) if
µ(Ωx0,t) ≥ tDµ(Ω), for t ∈ [0, 1]. (4)
this is a global estimate!
Riemannian with Ric ≥ 0 satisfy MCP(0, n)
[Julliet, 2009] the Heisenberg group satisfies MCP(0, 5)
[Rifford, 2013] Any ideal Carnot group satisfies MCP(0,D) for some integerD ≥ Q + n − k .
It is consistent with our theorem since N ≥ Q + n − k .
Conjecture: Carnot groups satisfy MCP(0,N )?
For Step 2 free Carnot groups Q + n− k ∼ k2 and N ∼ k3
at the moment → MCP(0,14) for the (3,6) Carnot group. (w/ L.Rizzi)
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 36 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Free step 2 Carnot groups
These are Carnot groups with growth vector (k , n) with
n = k +k(k − 1)
2
The generic geodesic is ample and equiregular with geodesic growth vector:
k
k − 1
2
1
k2
(k − 1)2
12
→ spec(Iλ) = 12, 22, . . . , k2 and N = k(k+1)(2k+1)6
∼ k3
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 37 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
Final remarks
→ Final remarks on MCP
What about non nilpotent situations?
Higher order terms in µ(Ωx0,t)? Bounds on curvature?
→ Open questions
What about other kind of comparison? (Laplacian eigenvalues, etc.)
Comparison for solution of heat equation? (the function ct appears in theheat kernel expression)
pt(x , y) =1
tn/2exp
(
−d2(x , y)
4t
)
(C + O(t))
study of the heat kernel along geodesics
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 38 / 38
Motivation Affine control systems and geodesic cost Geodesic growth vector, ample geodesics and curvature Applications: Laplacian
THANK YOU FOR YOUR ATTENTION!
Davide Barilari (Univ. Paris Diderot) SR Curvature October 2, 2014 38 / 38
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