Notes of Alessandro Ottazzi’s lecture nr 3

Posted on May 11, 2012 by

1. Liouville’s theorem

Let {f} be a {C^2} conformal map on an open domain of {{\mathbb R}^n}, {n\geq 3}. Observe that {f} is conformal if and only if at every point {x}, {Df(x)} belongs to the conformal group {CO(n)=\{A\in Gl(n,{\mathbb R})\,;\,\exists \mu>0} such that {AA^{\top}=\mu I\}}. In other words, {f} is an automorphism of the constant {G}-structure, {G=CO(n)}.

Infinitesimal conformal transformations are of the form

\displaystyle \begin{array}{rcl} V=\sum_{i=1}^{n}v_i \frac{\partial}{\partial x_i}, \end{array}

where the matrix {(\frac{\partial v_i}{\partial x_j}} belongs to the Lie algebra

\displaystyle \begin{array}{rcl} \mathfrak{co}(n)=\{X\in Gl(n,{\mathbb R})\,;\,\exists \nu,\,X+X^{\top}=\nu I\}. \end{array}

1.1. First step : prolongation and infinitesimal transformations

The prolongation is of the form {{\mathbb R}^n \oplus \mathfrak{co}(n)\oplus\mathcal{G}^{(1)}(\mathfrak{co}(n))}, it is isomorphic, as a Lie algebra, to {\mathfrak{so}(1,n+1)}. This is a classical computation. This is a Liouville theorem for vectorfields.

1.2. Second step : conformal transformations

Assume that {f(0)=0}. Let {U} an infinitesimal conformal transformations. So is {f_* U}, but, a priori, {f_* U} is only {C^1}. Mollify it. Since the equations are linear with constant coefficients, the mollified vectorfield is still conformal, thus belongs to {\mathfrak{so}(1,n+1)}. As a limit of such, {f_* U \in\mathfrak{so}(1,n+1)}. Whence a group homomorphism {\Phi:Conf\rightarrow Aut(\mathfrak{so}(1,n+1))=O(n,1)}.

One show that this homomorphism is injective. Indeed, assume that {g(0)=0} and {g_* U=f_* U} for all {U\in \mathfrak{so}(1,n+1)}. Then {g^{-1}\circ f} maps a constant vectorfield to itself, its differential equals identity, it is a translation, therefore it equals identity.

On the Möbius group {O(n,1)}, {\Phi} is identity, thus showing that {f\in O(n,1)}.

2. Tanaka prolongation theory

Let {G\subset Gl(n,{\mathbb R})} be a Lie group. Assume {G} leaves a vectorsubspace {V\subset {\mathbb R}^n} invariant. A {G}-structure subsumes a plane distribution. The Singer-Sternberg prolongation works if the {G}-structure is integrable, which often forces the distribution to be involutive, and we do not want this.

Tanaka replaces constant {G}-structures by different models: left-invariant {G}-structures on stratified Lie groups.

Definition 1 A model space (for Tanaka) is a stratified Lie algebra

\displaystyle \begin{array}{rcl} \mathfrak{n}=\mathfrak{g}_{-s}\oplus\cdots\oplus\mathfrak{g}_{-1}. \end{array}

such that for all {-s+1\leq j\leq -1},

\displaystyle \begin{array}{rcl} [\mathfrak{g}_{-1},\mathfrak{g}_{j}]=\mathfrak{g}_{j-1}. \end{array}

The corresponding Lie group {N} comes equipped with a horizontal distribution {\mathcal{H}N}. Contact mappings (mappings that preserve {\mathcal{H}N}) can be viewed as automorphisms of a geometric structure.

Definition 2 An infinitesimal automorphism is a vectorfield whose local flow consists of contact transformations.

Remark 1 {V} is an infinitesimal contact transformation if and only if {[V,\Gamma(\mathcal{H}N)]\subset\Gamma(\mathcal{H}N)}.

2.1. The full prolongation

We first define the full prolongation, i.e. Lie algebra relevant to the determination of all infinitesimal contact transformations.

Definition 3 {Prol(\mathfrak{n})=\bigoplus_{k=-s}^{\infty}\mathfrak{g}_{k}(\mathfrak{n})}, where {\mathfrak{g}_{0}(\mathfrak{n})=Der_{0}(\mathfrak{n})} is the space of degree {0} derivations, and for positive {k},

\displaystyle \begin{array}{rcl} \mathfrak{g}_{k}=\{u:\mathfrak{n}\rightarrow \mathfrak{n}\oplus\mathfrak{g}_0 (\mathfrak{n})+\cdots+\mathfrak{g}_{k-1} (\mathfrak{n})\,;\,\forall X,\,Y\in\mathfrak{n},\,u[X,Y]=u(X)(Y)-u(Y)(X)\}. \end{array}

Note that if {\mathfrak{g}_{k-1}(\mathfrak{n})=0}, then {\mathfrak{g}_{k}(\mathfrak{n})=0}.

Definition 4 The Lie bracket structure is defined as follows. For {u\in\mathfrak{g}_{k}(\mathfrak{n})}, {u'\in\mathfrak{g}_{k'}(\mathfrak{n})} and {X\in\mathfrak{n}}, set {[u,X]=(X)} and inductively

\displaystyle \begin{array}{rcl} [u,u'](X):=[u,[u',X]]-[u',[u,X]]. \end{array}

Definition 5 Replacing {Der_{0}(\mathfrak{n})} by a subalgebra {\mathfrak{g}_0 \subset Der_0(\mathfrak{n})}, one gets {Prol(\mathfrak{n},\mathfrak{g}_0)}.

2.2. Finiteness criterion

Definition 6 Define

\displaystyle \begin{array}{rcl} \mathcal{G}^{(0)}=\{u\in\mathfrak{g}_{0}\,;\,\forall X,\,Y\in\mathfrak{n},\, u[X,Y]=0 \}. \end{array}

Note {\mathcal{G}^{(0)}} can be viewed as a subalgebra of {\mathfrak{gl}(\mathfrak{g}_{-1})}, and thus admits a prolongation in the sense of Singer and Sternberg.

Theorem 7 (Tanaka) {Prol(\mathfrak{n},\mathfrak{g}_0)} is finite dimensional if and only if {\mathcal{G}^{(0)}} is of finite type in the sense of Singer and Sternberg.

2.3. Examples

Example 1 Let {N} be Heisenberg group. For the full prolongation, {\mathcal{G}^{(0)}=\mathfrak{sl}(2,{\mathbb R})} contains rank one matrices, so the full prolongation is infinite dimensional.

Example 2 Let {N} be Heisenberg group. Let {\mathfrak{g}_0 =\{u\in Der_0 (\mathfrak{n})} whose restriction to {\mathfrak{g}_{-1}} belongs to {\mathfrak{co}(2)}. Then {\mathcal{G}^{(0)}=\mathfrak{so}(2)}, so the prolongation is finite dimensional.

This implies that infinitesimal conformal contact transformations form a finite dimensional Lie algebra, but we have not proved it yet. The prolongation can be computed to be

\displaystyle \begin{array}{rcl} Prol(\mathfrak{n},\mathfrak{g}_0)=\mathfrak{n}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2 =\mathfrak{su}(2,1). \end{array}

Example 3 Let {N} be the group of unipotent upper triangular {4\times 4}-matrices. Then the full prolongation is

\displaystyle \begin{array}{rcl} Prol(\mathfrak{n})=\mathfrak{n}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2\oplus\mathfrak{g}_3. \end{array}

Whereas {\mathcal{G}^{(0)}(\mathfrak{n})} vanishes.

2.4. Prolongation of contact vectorfields

We are aiming at relating the Lie algebra of infinitesimal contact transformations to the full prolongation.

Definition 8 View a vectorfield on {N} as a {\mathfrak{n}}-valued function on {N}, {A_U : N\rightarrow\mathfrak{n}}.

Note that {A_{[U,W]}=[A_U,A_V]+U.A_W-W.A_U}.

If {U} is a contact vectorfield and {W} is horizontal and left-invariant, then {[U,W]} is horizontal again, and

\displaystyle \begin{array}{rcl} A_{[U,W]}=[A_U,W]-W.A_U. \end{array}

This implies that the right hand side is horizontal, i.e., for all {m\geq 2},

\displaystyle \begin{array}{rcl} W.A_{U}^{-m}=[A_U^{-m+1},W]. \end{array}

This set of equations characterizes infinitesimal contact transformations.

Example 4 Let {N} be the Heisenberg group. Then {V=fX+gY+hT} is a contact vectorfield if and only if {Xh=-g} and {Yh=f}.

Indeed, in this example,

\displaystyle \begin{array}{rcl} A_{V}^{-1}=(f,g,0), \quad A_{V}^{-2}=(0,0,h), \end{array}

\displaystyle \begin{array}{rcl} Xh=X.A_{V}^{-2}=[A_{V}^{-1},X]=-(0,0,g), \quad \end{array}

Iterating the equation above gives, for all {m>r}, and left-invariant {W\in\mathfrak{g}_{-r}},

\displaystyle \begin{array}{rcl} W_{-r}.A_{U}^{-m}=[A_U^{-m+1},W_{-r}]. \end{array}

Proposition 9 Let {U} be a vectorfield. The expression

\displaystyle \begin{array}{rcl} A_U^0 (W_{-r})=W_{-r}.A_{U}^{-r},\quad W_{-r}\in\mathfrak{g}_{-r}, \end{array}

defines a degree {0} derivation of {\mathfrak{n}}. Therefore, it defines a map {A_U^0 : N\rightarrow \mathfrak{g}_0}.

Example 5 Heisenberg again. Then

\displaystyle \begin{array}{rcl} A_V^0 =\begin{pmatrix} Xf &Yf&0 \\ Xg &Yg&0\\ 0&0&Th \end{pmatrix}. \end{array}

The contact equations are expressible in terms of {A_V}, {A_V^0 \in\mathfrak{g}}, {A_V^1}, …. which are higher and higher derivatives. This provides us with a map from infinitesimal contact transformations to the full prolongation. One shows that this map is an isomorphism.

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Notes of Alessandro Ottazzi’s lecture nr 2

Posted on May 9, 2012 by

Quasiconformal maps on Carnot groups

This survey is a break in the mini-course. It contains results of a metric nature (quasiconformality belongs to metric geometry).

1. Inspiring classical results

Theorem 1 (Liouville 1850) Every smooth conformal map of an open subset of {{\mathbb R}^n}, {n\geq 3}, is the restriction of a Möbius transformation.

A Möbius transformation belongs to the group generated by translations, rotations and inversions. This is a finite dimensional Lie group. So Liouville’s theorem is a rigidity result : conformal mappings are scarce in higher dimension. In contrast, the Riemann mapping theorem shows that conformal mappings are flexible in 2 dimensions.

Theorem 2 (Riemann 1850, Poincaré, Koebe 1907) Every proper simply connected open subset of {{\mathbb R}^2} is conformally equivalent to the disk.

To recover a bit of flexibility, quasiconformal mappings have been introduced.

Definition 3 (Grötzsch 1928) A homeomorphism {f} between Euclidean domains is {K}-quasiconformal if for any point {p}, the {r}-ball at {p} is taken to a domain containing an {h(p,r)} ball and contained in an {L(p,r)} ball, with {\limsup_{r\rightarrow 0} L/r\leq K}.

Theorem 4 (Ahlfors, Väisälä, Gehring) Liouville’s theorem extends to {1}-quasiconformal mappings.

In other words, the differentiability assumption in Liouville’s theorem can be removed.

One may wonder wether quasiconformal mappings are somewhat differentiable anyway. The answer is yes. They are absolutely continuous on lines, they admit distributional derivatives belonging to {L_{loc}^q} for some {q>n}, they are almost everywhere differentiable. But not much more in general.

2. SubRiemannian setting

Grötzsch’ definition is purely metric so it makes sense for subRiemannian manifolds. Interest in the subject in subRiemannian geometry started with the following observation.

Theorem 5 (Koranyi, Reimann 1985) The Heisenberg group admits non Möbius quasiconformal mappings.

In fact, they construct an infinite dimensional space of smooth vectorfields whose flows consist of quasiconformal mappings. They check that smooth contact transformations are locally quasiconformal. They use Paulette Libermann’s parametrization of infinitesimal contact transformations by functions

2.1. Rigidity

Definition 6 Let {\mathbb{G}} be a Carnot group. Say that {\mathbb{G}} is rigid if local contact mappings form a finite dimensional Lie group.

Example 1 Euclidean spaces and Heisenberg groups are not rigid.

Later, people have accumulated rigidity results.

Theorem 7 (Yamaguchi, Pansu, Cowling, De ?, Koranyi, Reimann, Ricci, Ottazzi, Warhurst,…) {H}-type groups, Iwasawa unipotent factors, free nilpotent Lie groups, Hessenberg manifolds… are rigid.

Until recently, only jet spaces had been shown to be non rigid (Warhurst 2005).

Theorem 8 (Ottazzi 2008) Let {\mathbb{G}} be a Carnot group. Assume that there exists a horizontal vector {X} such that {ad_X} has rank {\leq 1}. Then {\mathrm{G}} is non rigid.

Example 2 Engel’s group is non rigid.

2.2. Towards a characterization of rigid Carnot groups ?

The converse of Theorem 7 is not far from being true. First, go to the complexified Lie algebra {\mathfrak{g}\otimes\mathbb{C}}. Second, there remains a regularity issue to solve.

Theorem 9 (Ottazzi, Warhurst) Let {\mathbb{G}} be a Carnot group. Assume that there exists a complex horizontal vector {X} such that {ad_X} has rank {\leq 1}. Then {\mathrm{G}} is non rigid.

Conversely, assume that there exist no complex horizontal vectors {X} such that {ad_X} has rank {\leq 1}. Then quasiconformal mappings of class {C^2} form a finite dimensional Lie group.

Theorem 9 is related to a classical fact for {G}-structures. Indeed, there exists a complex horizontal vector {X} such that {ad_X} has rank {\leq 1} if and only if {\mathfrak{g}} has a rank {\leq 1} graded derivation which vanishes on commutators.

2.3. Generalized Liouville theorem

Theorem 10 (Ottazzi, Warhurst) Let {\mathbb{G}} be a Carnot group different from {{\mathbb R}^2}. Then {1}-quasiconformal maps on {\mathrm{G}} form a finite dimensional Lie group.

We rely on a recent regularity theorem.

Theorem 11 (Capogna, Cowling 2006) Let {\mathbb{G}} be a Carnot group. Then {1}-quasiconformal maps on {\mathrm{G}} are smooth.

The proof goes in two steps.

  1. Show that infinitesimal conformal mappings form a finite dimensional Lie algebra. This relies on Tanaka prolongation.
  2. A smooth conformal mapping defines an automorphism of this algebra. Show that the automorphism uniquely determines the mapping up to a left translation.

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Notes of Alessandro Ottazzi’s lecture nr &

Posted on May 8, 2012 by

Mini course on Tanaka prolongation

Extends Singer and Sternberg’s theory (late sixties). Its goal is to understand the equivalence problem and deformations of {G}-structures. In the seventies, Tanaka adapted it to subRiemannian geometry. Nowadays, it can be used to solve problems in subRiemannian geometry like

  • Determine the isometry group of a subRiemannian manifold.
  • Determine the conformal group of a subRiemannian manifold.
  • Guess differential invariants of geometric structures on plane distributions.

We shall explain the theory, starting from the classical case, and show some applications.

Plan

  1. {G}-structures
  2. Singer-Sternberg prolongation
  3. The Liouville theorem for conformal maps of {{\mathbb R}^n}
  4. Tanaka prolongation theory
  5. Liouville theorem for Carnot groups

Reference for the first 3 sections: Kobayashi’s book Transformation groups. The original paper by Singer and Sternberg contains a historical overview of the problem.

1. {G}-structures

Let {M} be a smooth {n}-manifold. Let {G} be a Lie group.

Definition 1 A {G}-principal bundle over {M} is a manifold {P} with a free right {G}-actions and a submersion {P\rightarrow M} which induces a diffeomorphism {P/G \simeq M}.

Definition 2 The frame bundle {LM} over {M} is the union of {L_m M}, {m\in M}, where {L_m M} is the set of ordered bases of {T_m M}. It admits a manifold structure which turns it into a {Gl(n,{\mathbb R})}-principal bundle.

Definition 3 Let {G\subset Gl(n,{\mathbb R})} be a closed subgroup. A {G}-structure on {M} is a {G}-sub-bundle of the frame bundle {LM} over {M}, i.e. {P\subset LM}, and for {p\in P} and {A\in Gl(n,{\mathbb R})}, {pA\in P \Leftrightarrow A\in G}.

Definition 4 A {G}-structure {P} on {M} is integrable if locally there exist local coordinates whose frames belongs to {P}.

Example 1 If {G} is trivial, a {\{1\}}-structure is a global frame. Such a structure exists if and only if {M} is parallelizable.

Definition 5 Let {P} be a {G}-structure on {M}. An automorphism of {P} is a diffeomorphism {f:M\rightarrow M} such that the induced tangent map {Tf(P)\subset P}. Equivalently, when expressed in frames taken from {P}, the differential of {f} at each point belongs to {G}.

Definition 6 Let {P} be a {G}-structure on {M}. An infinitesimal automorphism of {P} is a vectorfield whose local flow is made of automorphisms.

Remark 1 If the {G}-structure is integrable, a vectorfield {V} is an infinitesimal automorphism if and only if, in adapted coordinates, the differential of {V} belongs to the Lie algebra {\mathfrak{g}}.

Example 2 An {O(n)}-structure is the same as a Riemannian metric. Automorphisms are isometries, infinitesimal automorphisms are called Killing fields. An {O(n)}-structure is integrable if and only if it is flat.

Example 3 An {Sl(n,{\mathbb R})}-structure is the same as a choice of orientation and volume form. Automorphisms are volume and orientation preserving diffeomorphisms. Infinitesimal automorphisms are divergence free vectorfields. Every {Sl(n,{\mathbb R})}-structure is integrable.

This is an example of a {G}-structure of infinite type: automorphisms groupes are infinite dimensional.

2. Singer and Sternberg theory

We work locally, in {{\mathbb R}^n}, and with the flat {G}-structure. Let {V=\sum_{i=1}^{n}v_i \frac{\partial}{\partial x_i}} be a vectorfield. We want to characterize infinitesimal automorphisms in terms of their Taylor expansions. Write

\displaystyle \begin{array}{rcl} v_i(x)=v_i(0)+\sum_{k=1}^{\infty}\frac{1}{k!}a^{i}_{j_1,...,j_k}x_{j_1}\cdots x_{j_k}, \end{array}

where

\displaystyle \begin{array}{rcl} a^{i}_{j_1,...,j_k}=\frac{\partial^k}{\partial x_{j_1}...\partial x_{j_k}}(0). \end{array}

We note that

  • The {a^{i}_{j_1,...,j_k}} are symmetric in the lower indices.
  • The matrix {(\frac{\partial v^i}{\partial x_j})_{ij}\in\mathfrak{g}}.
  • Matrices {(a^{i}_{j_1...j_k})_{ij_1}\in\mathfrak{g}}.

2.1. Formal definition of Singer-Sternberg prolongation

Definition 7 For {k\in{\mathbb N}}, let {\mathcal{G}^{(k)}(\mathfrak{g})} denote the vectorspace of symmetric multilinear maps {T:{\mathbb R}^n \times\cdots\times{\mathbb R}^n \rightarrow {\mathbb R}^n} such that for all {v_1,\ldots,v_k\in{\mathbb R}^n}, {v\mapsto T(v,v_1,\ldots,v_k)\in\mathfrak{g}}.

In fact, it is an inductive definition: {T\in \mathcal{G}^{(k)}(\mathfrak{g})} if and only if for all {v\in{\mathbb R}^n}, {(v_1,\ldots,v_k)\mapsto T(v,v_1,\ldots,v_k)\in\mathcal{G}^{(k-1)}(\mathfrak{g})}, and {T} is symmetric.

Definition 8 {\mathfrak{g}} is of type {k} if {k} is the smallest integer such that {\mathcal{G}^{(k)}(\mathfrak{g})=0}. It is of infinite type if no such {k} exists.

2.2. The Lie algebra of jets

Denote by {\mathcal{G}^{(-1)}(\mathfrak{g})={\mathbb R}^n}. We define a graded Lie algebra structure on {\bigoplus_{k\geq -1}\mathcal{G}^{(k)}(\mathfrak{g})} which reflects the Lie algebra structure on the space of infinitesimal automorphisms of the integrable {G}-structure {P}. If {T\in \mathcal{G}^{(k)}(\mathfrak{g})} and {T\in\mathcal{G}^{(k')}(\mathfrak{g})}, let

\displaystyle \begin{array}{rcl} [T,T'](v_0,\ldots,v_{k+k'})&=&\frac{1}{k!(k'+1)!}\sum T(T'(v_{j_1},\ldots,v_{j_k'}),v_{j_{k'+1}},\ldots,v_{j_{k+k'}})\\ &&-\frac{1}{(k+1)!k'!}\sum T'(T(v_{j_1},\ldots,v_{j_k}),v_{j_{k+1}},\ldots,v_{j_{k+k'}}), \end{array}

where one sums over permutations. If {T\in\mathcal{G}^{(k)}(\mathfrak{g})} and {v\in\mathcal{G}^{(-1)}(\mathfrak{g})={\mathbb R}^n},

\displaystyle \begin{array}{rcl} [T,v](v_1,\ldots,v_k)=T(v,v_1,\ldots,v_k). \end{array}

By construction, the polynomial vectorfields {V=T(x,\ldots,x)} and {V'=T'(x,\ldots,x)} are infinitesimal automorphisms, their bracket is {[V,V']=[T,T'](x,\ldots,x)}.

2.3. Examples

Example 4 {G=O(n)}. Then the prolongation is {{\mathbb R}^n\oplus\mathfrak{so}(n)}.

Indeed, {\mathcal{G}^{(0)}(\mathfrak{so}(n))} consists of skew symmetric matrices, {\mathcal{G}^{(1)}(\mathfrak{so}(n))} consists of {(2,1)}-tensors which are symmetric in two indices and skew-symmetric in two others. A classical lemma asserts that such tensors vanish. Indeed,

\displaystyle \begin{array}{rcl} T^{i}_{jk}=-T^{k}_{ji}=-T^{k}_{ij}=T^{j}_{ik}=T^{j}_{ki}=-T^{i}_{kj}=-T^{i}_{jk}. \end{array}

which implies that {T=0}.

Example 5 {G=Sl(n,{\mathbb R})}. Then the prolongation is infinite dimensional.

Indeed, it is always so for Lie algebras which contain a matrix of rank {1}. To prove this, assume {A=v\otimes w^* \in\mathfrak{g}}. Let {k>0}. Set

\displaystyle \begin{array}{rcl} T(v_1,\ldots,v_{k+1})=w(v_1)\ldots w(v_{k+1})v. \end{array}

Moreover,

\displaystyle \begin{array}{rcl} z\mapsto T(v_1,\ldots,v_{k},z)=w(v_1)\ldots w(v_{k})Az \end{array}

belongs to {\mathfrak{g}}. Thus {T\in \mathcal{G}^{(k)}(\mathfrak{g})}.

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Slides of Mario Sigalotti’s talk

Posted on May 3, 2012 by

Yesterday, I missed Mario Sigalotti’s talk on Gauss-Bonnet for almost Riemannian surfaces.

Here are his slides.

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Notes of Ugo Boscain’s lecture

Posted on April 4, 2012 by

A model of human vision based on sub-Riemannian geometry

We look for an explanation for certain visual illusions and for the capacity of human brain to fill in the gaps in images.

The anthropomorphic algorithm we propose is not (yet) more efficient that more standard methods (segmentation, wavelets,…).

Goes back to Hoffman, rediscovered by Petitot, refined by Citti and Sarti, Agrachev, Charlot-Gauthier-Rossi. I will also present results by Yuri Sachkov, discovered indepedently by Duits.

1. From neurophysiology to the model

1.1. Neurophysiology

In the V1 visual cortex, neurons are sensible to both position and direction. So the brain storeds an image as a set of points and directions, i.e. a subset of {PT{\mathbb R}^2}.

An image is reconstructed by minimizing the energy necessary to excite groups of neurons that are not excited by the image in {PT{\mathbb R}^2}.

Huber and Wiesel (Nobel prize 1981) observed that neurons split into groups each of which is sensitive to a specific direction. These groups are called orientation columns. They are in turn grouped into hypercolumns. There are both horizontal and vertical connections between orientation columns.

1.2. Geometry of {PT{\mathbb R}^2}

Every {C^1} curve in {{\mathbb R}^2} has a canonical lift. The lifts of plane curves are in 1-1 correspondance with {C^1} curves tangent to a contact structure on {PR{\mathbb R}^2}.

1.3. How V1 reconstructs an interrupted curve

By minimizing something. What ? When moving objects with the hands, the brain minimizes a compromise between energy and the stress of muscles (external cost). For reconstruction of images, the (internal) minimized cost is the energy necessary to activate neurons that are not naturally activated by the image of the interrupted curve. Given a neuron that is already active, it is easy to activate nearby neurons, where proximity in measured in brain geometry, i.e. by connections in hypercolumns.

The simplest cost is the Riemannian length of curves which are lifts of planar {C^1} curves. This leads to a problem in sub-Riemannian geometry on a Lie group, the roto-translation group of the plane. There is only one left-invariant cost, up to planar dilations (Agrachev). One can replace length by energy (integral of squared derivative).

1.4. Advantages of this cost

In terms of planar geometry, this provides a compromise between length and curvature of the planar curve.

Minimizers exist in the function space of absolutely continuous horizontal curves.

There is a natural hypo-elliptic diffusion semi-group, which can be used for reconstruction.

1.5. Drawbacks of this cost

There are minimizers with cusps, which are not observed in psychological experiments.

1.6. The Citti-Sarti proposition to avoid cusps

Consider only curves in {PT{\mathbb R}^2} whose speed has always non-vanishing {x} component. One looses existence of minimizers (observed by all the authors above). Minimizing sequences tend to converge to curves which do not statisfy the boundary condition any more.

Theorem 1 Classification of the boundary conditions for which existence of minimizers exist.

2. Reconstruction of curves

2.1. Computation of Sub-Riemannian distance

The Hamiltonian flow is closely connected to the phase portrait of the integrable pendulum. Closed trajectories of the pendulum correspond to fronts (alternating cusps). Other trajectories correspond to kinds of cycloids. The issue of optimality of trajectories is hard (Sachkov and Moisseev).

The cut locus of a point is genuinely {2}-dimensional, unlike for the Heisenberg left-invariant metric.

2.2. Preliminary reconstruction results by Sachkov

Sachkov took an image made of level sets of a function, corrupted it by rubbing out big oval parts. His reconstruction manages to connect level sets corresponding ot the same level. How ?

2.3. Sachkov’s algorithm

Input is the image of level sets of a function.

1. Smooth the image with a Gaussian to get a Morse function (physiologists argue that such a smoothing indeed occurs in the human eye).

2. Lift the image to {PT{\mathbb R}^2}, get a smooth surface (resolution of singularities by blow-up), view it as a generalized function on {PT{\mathbb R}^2}. Use it as an initial condition for the hypo-elliptic heat equation (i.e. perform convolution with the heat kernel).

3. Project down onto {{\mathbb R}^2} by integrating along fibers. Output.

2.4. Underlying mathematical results

Theorem 2 Explicit expressions for the sub-Riemannian heat kernels on low dimensional Lie groups.

2.5. Numerical implementation

Attemps by Gauthier. There does not seem to be convergence results for the numerical computation of the hypo-elliptic diffusion. Methods like finite elements do not apply due to non-commutativity. Gauthier used Fourier analysis and the expression of the heat kernel.

The images produced look very good (like a hand-drawn picture), even when a large part of the original image was corrupted, but computation is very expensive at present.

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Slides of Mauricio Godoy’s talk

Posted on April 4, 2012 by

Mauricio Godoy spoke on

SubRiemannian structures on odd dimensional spheres

on march28th, 2012.

Here are his slides: Godoy_IHP2012

(unfortunately, I missed his talk).

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Notes of Davide Barilari’s lecture

Posted on March 14, 2012 by

Geometry and heat equation on sub-Riemannian manifolds

Which is the right Laplace operator associated to a sub-Riemannian metric ? Which of its properties can be read from the Carnot-Carathéodory distance ?

1. Intrinsic volume and sub-Laplacian

These were defined by Brockett (1982) in dimension 3, later by Popp for step 2 equiregular distributions, by Montgomery in general in 2002. Let us start with volume.

In dimension 3, pick an orthonormal basis {(X_1,X_2)} of the distribution. Then

\displaystyle (X_1,X_2,[X_1,X_2])

is a basis. Take the dual basis of differential 1-forms, wedge them. The result does not depend on the choice of basis (Brackett).

For a 2-step equiregular distribution, equip {TM/\Delta} with the inner product induced by the Lie bracket, which is canonically defined surjective linear map. Pick a complement to {\Delta} in {TM}, get a Riemannian metric. It turns out that its volume element does not depend on the choice of complement (Popp).

Once one has an intrinsic volume, one gets an intrinsic divergence operator.

Definition 1 (Montgomery) The sub-Laplacian {L} is defined by {L=div \circ grad} where {grad} is the horizontal gradient.

2. Intrinsic volume versus Hausdorff measure

2.1. How does volume relate to Hausdorff measure ?

Theorem 2 (Agrachev, Barilari, Boscain) The density {f} of Hausdorff spherical measure with respect to Popp volume is continuous.

It is constant in dimension {\leq 4}, or in dimension {5} and corank {1}.

In higher dimensions, corank {1}, it is not smooth (not {C^5}).

Gauthier showed that {f} is generically {C^1} in dimensions {(4,6)}.

2.2. Proof

{f} is the volume of unit ball in the nilpotent approximation. In low dimensions, the nilpotent approximation is constant. In dimension {\geq 5}, the unit ball, specifically, the cut time, is not a smooth function of the Lie algebra structure.

If cut time coincides with conjugate time, we gain a bit of regularity. Indeed,

\displaystyle \begin{array}{rcl} vol(B)=\int_{\{\lambda\}}\int_{0}^{cut~time(\lambda)}|det(d \exp)|\,dt\,d\lambda. \end{array}

By definition, integrand {det(d \exp)} vanishes at conjugate time.

3. Heat equation

3.1. Classical results

Hörmander : {L} is hypo-elliptic. Therefore, heat flow is well defined with a smooth kernel {p_t}.

Theorem 3 (Léandre)

\displaystyle \begin{array}{rcl} \lim_{t\rightarrow 0}4t p_t(x,y)=-d(x,y)^2. \end{array}

This asymptotic can be refined at points where the distance is smooth.

Theorem 4 (Agrachev) Squared distance to a point {x_0} is smooth away from the cut-locus, i.e. on the open and dense set {\Sigma(x_0)} of points joined to {x_0} by a unique normal non-conjugate minimizer. On that set, the differential of {d^2/2} is the normal extremal of that minimizer (a covector) at the endpoint.

Theorem 5 (Bénarous) If {y\in \Sigma(x)},

\displaystyle \begin{array}{rcl} p_t(x,y)\sim t^{-n/2}e^{-\frac{d(x,y)^2}{4t}}. \end{array}

3.2. Expansion at cut points

At cut points, one can still say something. The idea is to use the semi-group property,

\displaystyle \begin{array}{rcl} p_{t}(x,y)=\int_{M}p_{t/2}(x,z)p_{t/2}(z,y)\,dz, \end{array}

and to plug in Bénarous’ asymptotics, leading to

\displaystyle \begin{array}{rcl} p_{t}(x,y)&\sim&t^{-n}\int_{M}e^{-\frac{d(x,z)^2}{2t}}e^{-\frac{d(z,y)^2}{2t}}\,dz\\ &=&t^{-n}\int_{M}e^{-\frac{h_{x,y}(z)}{2t}}\,dz, \end{array}

where

\displaystyle h_{x,y}(z)=\frac{1}{2}(d(x,z)^2+d(y,z)^2).

is the hinge energy function. For the asymptotics when {t} tends to {0}, what matters is the behaviour of {h_{x,y}} in a neighborhood of its minima.

3.3. From estimates on hinge energy to asymptotics of {p_t}

Lemma 6 {h_{x,y}} achieves its minimum along the set of mid-points of minimal geodesics from {x} to {y}.

Theorem 7 Assume that {\gamma} is a strongly normal minimizer, i.e. all sub-arcs are strictly normal. Let {z_0} be its mid-point. Then

  1. {y} is conjugate to {x} along {\gamma} {\Leftrightarrow} {Hess_{z_0}h_{x,y}} is degenerate.

Theorem 8 Assume that all minimizers from {x} to {y} are strongly normal. Then

\displaystyle \begin{array}{rcl} p_t(x,y)=t^{-n}\int_{N}(c_{x,y}(z)+O(t))e^{-\frac{h_{x,y}(z)}{2t}}, \end{array}

where {N} is a neighborhood of the set of mid-points.

In particular, if there exist coordinates such that

\displaystyle \begin{array}{rcl} h_{x,y}(z)=\frac{1}{4}d^2(x,y)+z_1^{m_1}+\cdots+z_k^{m_k}+o(|z|^{m_k}), \end{array}

we get an expansion whose leading term in {t} is {t^{-n+\sum\frac{1}{m_j}}}.

For instance, if {h} has Morse minima, one recovers the asymptotic in {t^{-n/2}}.

Corollary 9 Up to constants,

\displaystyle \begin{array}{rcl} t^{-\frac{n}{2}}\leq e^{-\frac{d^2(x,y)}{4t}}p_t (x,y) \leq t^{-n+1}. \end{array}

We can improve this last result when there is a 1-parameter family of minimizers.

3.4. Examples

Example 1 Heisenberg group.

Using the explicit formula for {p_t}, one sees that along the vertical axis, {p_t(x,y)\sim t^{-2}\exp(-\pi z/t)}. The exponent {2} (instead of {3/2} at generic points) accounts for the fact that there is a {1}-parameter family of minimizers.

Example 2 Grushin plane.

One can compute the cut and conjugate loci of ordinary points. We have studied in detail the cut-conjugate points (there is only one up to symmetry). We have an expansion of {h} is a neighborhood, which leads to

\displaystyle \begin{array}{rcl} p_t \sim t^{-5/4}\exp(-\pi^2/t). \end{array}

We have been able to compute the Grushin plane merely because its geodesic flow is integrable in terms of trigonometric functions. The fact the metric degenerates along a line is not essential. We expect the result to generalize to all generic Riemannian metrics.

4. Questions

Is the density {f} always {C^1} ?

Expansion of diagonal heat kernel ? We have the expansion

\displaystyle \begin{array}{rcl} p_t (x,x)\sim t^{-2}(1+\kappa(x)t +O(t^2)) \end{array}

in dimension {3}. {\kappa} can be interpreted as curvature (vanishes for Heisenberg group). What about higher dimensions ?

Abnormal geodesics ? Not accessible by our technique.

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