Notes for Michele Stecconi’s Seminar at SISSA

Pfaffian Equations: equation on $TM$ of the kind $\{\omega=0\}$, where $\omega\in\Lambda M$.

We will discuss smooth distributions of plane fields $\xi_p\subset T_p M$ on  a $3$ dimensional manifold $M$. Locally we can describe them as a quotient $P_p \Lambda^1M:=\Lambda^1M/\sim p$, where we mean that $\omega_1\sim_p\omega_2$ if and only if there exists an open neighborhood $O_p$ of $p$ and a diffeomorphisms $\phi$ on this neighborhood such that $\omega_2=f\phi^*\omega_1$.

Local structure of the generic plane field

• Let $X$ be a topological space. We say that a subset $R\subset X$ is residual if and only if $R=\bigcap_{n} A_n$, where the sets $A_n$ have the property that $\overline{\textrm{int}(A_n)}=X$.
• A property is generic in $X$ if and only if it holds on a residual subset of $X$.
• Theorem (Thom) Let $\pi:E\to M$ be any vector bundle, $W\subset J^k(M\backslash E)$ for some $k\in\mathbb N$. Assume that $W$ is a submanifold; then the set $\{\sigma\in C^\infty(M\backslash E\,|\, j^k\sigma\pitchfork W)\}$ is residual.
• We recall that if $f:M\to N$ is a smooth map and $W\subset N$ is a submanifold, we say that $f\pitchfork W$ if and only if $\forall x\in f^{\leftarrow}(W)$ we have $f_*(T_x M)+T_{f(x)}W=T_{f(x)}N$.
• Corollary: For the generic section $\sigma\in C^\infty(M\backslash E)$, the set $\{p\in M\,|\, j^k\sigma(p)\in W\}$ is a regular submanifold of codimension equal to $\textrm{codim}(W)$

We apply this machinery to our case:

• if $\omega\wedge d\omega(p)\neq 0$, then we have a contact structure, and all contact structures are locally equivalent by Darboux theorem.
• On the contrary, we may define the set $W=\{[\omega]\in J^1(M\backslash T^*M)\,|\, [\omega]\wedge d[\omega]=0\}$, and see that it is a submanifold of codimension one. Pull it back on the manifold to obtain a surface, the Martinet surface, that is the set $\{p\in M\,|\,\omega\wedge d\omega(p)=0\}$ (regular for the generic section).
• In $\mathbb R^3$ the condition $\omega\wedge d\omega=h dx\wedge dy\wedge dz$, and the Martinet surface $\mathcal M$ is the zero set of $h$.
• Define a foliation tangent to $\omega\big|_{\mathcal M}$, that is on the set $\omega\wedge d\omega=0$ and $\omega|_{\mathcal M}=\omega\wedge dh\neq 0$.
• Claim: let $X$ be a smooth vector field on $\mathcal M$ such that $\omega(X)=0$ and $X=0$ if and only if $\omega\big|_{\mathcal M}=0$. Then, whenever $X_p=0$, in any local coordinate system we have $\textrm{tr}\left(\frac{\partial X(p)}{\partial x}\right)=0$. Indeed assume that $u,v$ are coordinates on $M$. Then $\omega\big|_{\mathcal M}=\alpha du+\beta dv$ and $X=f(\beta\partial_u-\alpha\partial_v)$. Moreover $0=(d\omega)(p)\big|_{\ker \omega}=(d\omega)(p)\big|_{\mathcal M}=d(\omega\big|_{\mathcal M})_p=f(\frac{\partial \beta}{\partial u}-\frac{\partial \alpha}{\partial v})du\wedge dv=f\nabla X du\wedge dv$.

Theorem (Zhitomirskii): let $\omega\in \Lambda^1M^3$  be a generic never vanishing one-form. Then there exist a regular surface $\mathcal M$ and two discrete sets $E,S\subset \mathcal M$ such that

1. if $p\not\in \mathcal M$, then $\omega$ is locally equivalent to $dz+xdy$
2. if $p\in \mathcal M\setminus (E\cup S)$ then $\omega$ is locally equivalent to $dz+x^2dy$
3. if $p\in E$ then $\omega$ is locally equivalent to $dz+(xz+\frac{x^3}{3}+xy^2+b x^3y^2)dy$, for some $b\in\mathbb R$
4. if $p\in S$ then $\omega$ is locally equivalent to $dz+(xz+x^2y+b x^3y^2)dy$, for some $b\in\mathbb R$

Notes for Alexander Medvedev’s seminar at SISSA

$(H,g)$

$v\in H_p\subset T_pM$

• Time-like vector fields $g(v,v)<0$,
• Space-like vector fields $g(v,v)>0$,
• Null or light-like vector fields $g(v,v)=0$,
• Non space-like vector fields $g(v,v)\leq 0$

Choose a basis $X_0,\dotso, X_n$. We say that two such bases are co-oriented if $g(X_0,\overline X_0)<0$ and $\textrm{det}(X_i,\overline X_j)>0$

Motivations: consider the following optimal control problem $\dot q=X_0+\sum_{i=1}^k u_i X_i$ on $C=\{u|\sum u_i^2\leq 1\}$, and consider the metric $g$ defined by $g(X_0,X_0)=-1$, $g(X_i,X_j)=\delta_{ij}$.

Consider then a non space-like future-direction curve $\gamma$ such that $g(\dot\gamma,\dot\gamma)\leq 0$, with $g(\dot\gamma, X_0)<0$, that is $\dot\gamma=v_0X_0+\sum v_i X_i$, with $-v_0^2+\sum v_i^2\leq 0$. Notice that if we set $u_i=v_i/v_0$ we recover the previous control problem.

We want to study equivalence classes of such problems. Generally what we would like to do is, given a frame $X_1,\dotso, X_n\in T\overline M$, and the structure constants $[X_i,X_j]=C_{ij}^k(p) X_k$, to write down some functional equations depending on $C, X_i(C), X_jX_i(C),\dotso$.

If $g(X_1,X_1)=-1$, $g(X_1,X_2)=0$, $g(X_2,X_2)=1$, with $X_3=[X_1,X_2], [X_3,X_1]=0, [X_3,X_2]=0$, then any other such frame satisfies $(\overline X_1,\overline X_2)=(\cosh\theta X_1+\sinh\theta X_2, \sinh\theta X_1+\cosh\theta X_2)$, and $\overline X_3=X_3$.

Then we construct a frame bundle for $(g,H)$, call it $\mathcal G=(p,\theta),p\in M, \theta\in\mathbb R$. In particular we fix the zero section in this frame bundle to be $X_1,X_2,X_3=[X_1,X_2]$.

Every lift of a given structure differs by an action of $SO^+(1,1)$, and we can identify $X$ with $S_{X*}(X)\in\mathcal G$. We prolong our base manifold by considering the action of $SO(1,1)$ on $\mathcal{G}$. The generator of this action is $X_u=\frac{\partial}{\partial \theta}$, and the commutators are $[X_u,X_1]=X_2, [X_u,X_2]=X_1, [X_u,X_3]=0$.

Then any frame in $\mathcal{G}$ which project down to $X_1,X_2$ are of the form $Y_1=X_1+f\partial_\theta, Y_2=X_2+g\partial_\theta$. We need to choose $f,g$ canonically. How to? Just kill some structure constants on $\mathcal G$.

Then one can shows that with this canonical choice of $f$ and $g$, $C=\mathfrak{sl}_2(\mathbb R)$ and find two invariants $K=\frac{c_{23}^1+c_{13}^2}{2}-X_1(c_{12}^2)-X_2(c_{12}^1)-(c_{12}^2)^2-(c_{12}^1)^2$, and $h =\left( \begin{array}{cc} c & \frac{c_{23}^1-c{13}^2}{2} \\ \frac{c_{13}^2-c{23}^1}{2} & -c\end{array}\right)$, which are basically the anti-symmetric and symmetric part of the modulus.

Some Topical Open Questions in SR Geometry, Part II

Notes:

Consider the tangent bundle $T^*M$ (the dimension of $M$ is equal to $n$) and a particular class of Hamiltonians:

$H(p,q)=\sum_{i=1}^k\langle p,f_i(q)\rangle^2+\langle p,f_0(q)\rangle+Q(q)$

Assumptions:

• $Q$ non negative quadratic form,
• $\textrm{Lie}_q\{f_i, [f_0,f_i],\dotso, [f_0,\dotso,[f_0,f_i],\dotso],\dotso]\}=T_qM$

Consider the hamiltonian vector field $\vec{h}(p,q)=\frac{\partial h}{\partial p}\frac{\partial}{\partial q}-\frac{\partial h}{\partial q}\frac{\partial}{\partial p}$, and its time-$t$ flow $e^{t\vec{h}}:T^*M\to T^*M$. If $\pi:T^*M\to M$ is the natural projection, we may consider the exponential map

$\mathcal{E}_h^t:T^*_{q_0}M\to M$, defined by $\mathcal{E}_h^t=\pi\circ e^{t\vec{h}}\big|_{T^*_{q_0}M}$

Question:

Is it true that there exist $p\in T^*_{q_0}M$ such that if we consider $\textrm{det}(d_p\mathcal{E}^t_h)\simeq O(t^N)$, for some $N$?

• If there exists such a $p$, we have in fact a Zariski open sets of covectors in $T^*_{q_0}M$ satisfying the same property. Indeed the question implies that there is maybe some high derivative of the determinant that does not vanish, which is a (terrible) rational function of $p$. If this is not identically zero, then it is non zero on a Zariski open set.
• The question can be substantially restated in terms of finding a system of privileged coordinates if we have a drift field.

Consider again the exponential map (in the homogeneous case, with no drift and potential) $\mathcal{E}_h^t:T_{q_0}^*M\to M$. We know from the first part that $\textrm{det}(d_p\mathcal{E}_h^t)=O(t^N)$.

We define a unimodular structure in this way: let $\mu\in \Lambda^n(M)$ be any volume-form, and consider $(\mathcal{E}^t_h)^*\mu\big|_p=t^Ng(t,p)\mu_{q_0}$, with $g(0,p)\not\equiv 0$. In the Riemannian case $g(t,p)=1+t^2\textrm{Ric}_p$, if we use the Riemannian volume form.

Define $\rho_{\mu}(p)=\frac{d}{dt}(\frac{1}{t^N}(\mathcal{E}_h^t)^*\mu_{q_0})\big|_{t=0}$. A structure is unimodular if there exists $\mu$ such that $\rho_\mu(p)=0$.

Question:

Characterize the unimodular structures.

Some topical open questions in SR geometry

Speaker: Andrei A. Agrachev (SISSA)

Venue: SISSA, Room A-136, February 9th – 4pm

Notes:

SubRiemannian structure: $M$ be a smooth manifold, $\Delta=\{\Delta_q\}_{q\in M}$ a distribution in $TM$ endowed with an inner product $g_q$, smoothly varying with respect to the base point.

Horizontal curves: $\gamma:[0,1]\to M$ such that $\dot\gamma(t)\in\Delta_{\gamma(t)}$ a.e. $t\in[0,1]$.

Energy: $J(q_0,q_1)=\inf_\gamma\{\int_0^1|\dot\gamma(t)|^2dt\,|\,\gamma(0)=q_0,\gamma(1)=q_1,\;\gamma\,\textrm{is an horizontal curve}\}$.

Assume that the distribution is locally given as $\Delta_q=\textrm{span}\{f_1(q),\dotso, f_k(q)\}$, with $g_q(f_i(q), f_j(q))=\delta_{ij}$; horizontal curves take the form $\dot\gamma(t)=\sum_{i=1}^k u_i(t) f_i(\gamma(t))$, with $\gamma(0)=q_0$ and $\gamma(1)=q_1$, while the energy becomes $J(u)=\frac{1}{2}\int_0^1|u(t)|^2dt$, where we mean that $u=(u_1,\dotso, u_k)\in L_2^k([0,1])$.

In the following we will assume that the Hormander condition holds, that is $\textrm{Lie}_q\{\Delta\}=T_qM$ for any $q\in M$; this in turn implies that the subriemannian distance between any two points is finite (Chow Rashevskii theorem), and the topology infuced by the subriemannian distance coincides with the standard manifold topology of $M$.

Endpoint map: The differential equation for horizontal curves induces a coordinate identification between controls and horizontal curves. We define, using this identification, $F_1: L^k_2([0,1])\to M$ given by $F_1(u)\mapsto \gamma_u(1)$.

More generally, we can consider the map $F_t(u)=\gamma_u(t)$. Then these maps are smooth. Our goal is to minimize $J|_{F^{-1}_1(q_1)}$

Remark: The space of horizontal curves joining two points is not a smooth manifold in general, indeed there exist abnormal curves (singular points for the Endpoint map), which is not the case in riemannian geometry. The most critical point is the constant curve, indeed as it is evident from the formula $D_0F_1 v=\int_0^1 \sum_{i=1}^k v_i(t)f_i(q_1)dt$, we can just obtain directions of $\Delta\subset TM$.

Subriemannian Hamiltonian: $H:T^*M\to \mathbb R$ defined by $H(p,q)=\frac{1}{2}\sum_{i=1}^k\langle p,f_i(q)\rangle^2$, where $p\in T^*_qM$. Integral curves of the hamiltonian satisfy $\dot p=-\frac{\partial h}{\partial q},\; \dot q=\frac{\partial h}{\partial p}$.

Remark: Normal geodesics are projections on $M$ of integral curves of $H$.

We evaluate the differential $D_u F_1[v]=\int_0^1 (P^1_t)_*\sum_{i=1}^k v_i(t) f_i(\gamma_u(t))dt$, where the flow $P$ is relative to the control $u$ (if $u$ where zero, it would have been the identity). Therefore $\textrm{conv}(D_uF)=\textrm{span}\{(P_t^1)_*\Delta_{\gamma(t)}\}_{t\in[0,1]}$. If we call $G_1(v)= P^0_1 F_1(v)$, then $\textrm{Im}_u G_1=\textrm{span}\{(P^0_t)_*\Delta_{\gamma(t)}\}$.

Flag: $\Delta^{(1)}_\gamma=\Delta_{q_0}$, $\Delta_\gamma^{(i)}=\frac{d^{i-1}}{dt^{i-1}}(P^0_t)_*\Delta_{\gamma(t)}\big|_{t=0}+\Delta^{(i-1)}_\gamma$. How to compute? Choose $X$ so that $X(q)\in\Delta_q$ for any $q\in M$ and $\dot\gamma(t)=X(\gamma(t))$, then $\Delta^i=\textrm{span}\{\underbrace{[X,[X,\dotso,\Delta]],\dotso]}_{\leq i-1}\}$.

We say that $\gamma$ is tame  if $\exists m>0$ such that $\Delta^{(m)}=T_{q_0}M$ (tame equals regular only if the distribution is analytic).

Theorem: Fix $q_0\in M$. There is a Zariski open subset in $T^*_{q_0}M$ such that any geodesics starting with $p_0$ in this subset is tame.

Question 1: What is the codimension of this set?

In terms of flag, we may compute the geodesic dimension of the subriemannian manifold. Try to contract a set $\Omega_1$ along geodesics: we obtain a family of sets $\Omega_t$ and $\textrm{vol}(\Omega_t)\simeq t^N\textrm{vol}(\Omega_1)$. Then $N$ is the geodesic dimension. This definition is well-posed just if there are no abnormal curves, since the Sard conjecture may still be not true.

Let $n_1=\textrm{dim}\Delta^{(1)}$, $n_i=\textrm{dim}\Delta^{(i)}-\textrm{dim}\Delta^{(i-1)}$. These numbers are constant on the complement of a Zariski closed set; it is called the growth number and do not depend on the geodesic. $N=\sum_{i=1}^m(2i-1) n_i$.

We may consider another flag $\Delta^1=\Delta$, $\Delta^i=[\Delta,\Delta^{i-1}]+\Delta^{i-1}$. The Hausdorff dimension is then computed as $N_h=\textrm{dim}\Delta^1+2(\textrm{dim}\Delta^2-\textrm{dim}\Delta^1)+3(\textrm{dim}\Delta^3-\textrm{dim}\Delta^2)+\dotso+$ and $N\neq N_h$

After this preliminary introduction we switch to a more general class of hamiltonians of the kind $h(p,q)=\sum_{i=1}^k\langle p,f_i(q)\rangle^2+\langle p,f_0(q)\rangle+Q(q)$, associated with a cost of the form $J(\gamma)=\int_0^1\frac{1}{2}|u(t)|^2-Q(\gamma(t))dt$ and a differential equation $\dot(\gamma(t))=f_0(\gamma(t))+\sum_{i=1}^k u_if_i(\gamma(t))$.

In this setting we have $\Delta=\textrm{span}\{f_1,\dotso,f_k\}$. The bracket generating property in this set is that $\textrm{Lie}\{f_i, [f_0,f_i]\}_{i=1}^k=T_{q}M$.

Question 2: Is it still true that we have a Zariski open set of tame geodesics?

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The Geometry of Random Lemniscates

Speaker: Antonio Lerario (SISSA)

Venue: SISSA, Room A-136, February 2nd – 4pm

The preimage of the unit circle under a rational map from the Riemann sphere to itself is called a lemniscate. Lemniscates appear in many branches of mathematics (from potential theory to algebraic geometry) and have been widely investigated. If we plot a “typical” lemniscate (for example sampling the coefficients of the polynomials defining the rational function from some Gaussian distribution) we see beautiful and mysterious pictures appearing (see the image below).

A random lemniscate of degree 400.

In this talk I will discuss a probabilistic approach to the study of these curves, presenting also a soft introduction to the methods of random algebraic geometry.

(This is joint work with E. Lundberg)

Switching in time-optimal problem. The 3D case with 2D control.

Speaker: Carolina Biolo (SISSA)

Venue: SISSA, Room A-137, 3rd of December – 14:00

We investigate the regularity of a time-optimal trajectory for an affine control system with drift in a $3$-dimensional manifold $M$, with a $2$-dimensional control $u=(u_1,u_2)$ with values in a closed unitary ball $U$. For this purpose,  we study the regularity of the correspondent time-optimal controls; each point of discontinuity of these controls is called switching. We prove that, when a particular generic condition on the vector fields of the control system is satisfied, every time-optimal trajectory is a concatenation of bang arcs, which are smooth, with isolated switching. We have found in which cases a time-optimal trajectory is just one bang arc, or could have switchings, that we are able to describe explicitly.

This is a joint work with A. A. Agrachev.

Edit: Here are the slides of the seminar provided by the author

Homotopy invisibility of Singular Curves

Speaker: Francesco Boarotto (SISSA)

Venue: SISSA, Room A-133, 25th of November – 14:00

In this talk we are going to show that for the generic sub-Riemannian structure we can perform an analogue of the classical Morse theory. In particular we will show that if there are no normal critical points of the extended endpoint map between two sublevel of the energy $E_1 and $\epsilon>0$ is a small parmeter, it is possible to deform any compact set in the space of controls having energy not greater than $E_2$ to a compact set having energy not greater than $E_1+\epsilon$. As a corollary, we will provide the counterpart of the classical min-max theorem for normal geodesics. If time permits we will discuss consequences and open problems related to our contruction.

This is a joint work with A. Lerario and A. A. Agrachev.

Edit: here are some additional slides to the talk