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?

Some topical open questions in SR geometry

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