Differential Properties of the Sub-Riemannian Distance Function



Marek Grochowski (Warsaw)


1. INTRODUCTION. It is well-known that if $(M,g)$ is a Riemannian manifold then the Riemannian distance from a fixed point $p_{0}$ is smooth on a set $U\backslash \{p_{0}\}$, where $U$ is a sufficiently small normal neighbourhood of $p_{0}$. The aim of this paper is to generalize this result over sub-Riemannian manifolds (theorems 1 and 2)1.

2. BASIC DEFINITIONS. Let $M$ be a smooth (i.e. of class $C^{\infty }$) connected $n$-dimensional manifold, $H$ a smooth $k$-dimensional distribution on it, $2\leq k<n$, and $g$ a smooth Riemannian metric on $H$. The triple $(M,H,g)$ is called a sub-Riemannian manifold.

We will deal only with a certain special class of curves, namely

DEFINITION 1   A curve $\gamma :[0,1]\longrightarrow M$ is called horizontal if it is absolutely continuous, has square integrable derivative with respect to some (and hence every) Riemannian metric on $M$, and $\dot{\gamma}(t)\in H_{\gamma (t)}$ a.e. on $[0,1]$

In the sequel we shall assume that $H$ satisfies Hörmander's condition which guarantees that any two points of $M$ can be joined by a horizontal curve ([LS]). Let $p\in M$ and denote by $\Omega _{p}$ the set of all horizontal curves starting from $p$. The set $\Omega _{p}$ carries the natural structure of Hilbert manifold (see [Bis]). Define the endpoint mapping $%% end_{p}:\Omega _{p}\longrightarrow M$: $end_{p}(\gamma )=\gamma (1)$. It can be proved that $end_{p}$ is of class $C^{\infty }$ ([Bis]).

DEFINITION 2   A curve $\gamma \in \Omega _{p}$ is called regular (resp. abnormal or singular) if the differential $d_{\gamma }end_{p}:T_{\gamma }\Omega _{p}\longrightarrow T_{\gamma (1)}M$ is surjective (resp. is not surjective).

For a given horizontal curve $\gamma :[\alpha ,\beta ]\longrightarrow M$ we define its length

$\displaystyle L(\gamma )=\int_{\alpha }^{\beta }g(v,v)^{1/2}dt.$     (1)

Now, we can define the sub-Riemannian distance

$\displaystyle d(p,q)=\inf \left\{ L(\gamma ):\;\gamma \in \Omega _{p,q}\right\} ,$     (2)

where $\Omega _{p,q}=end_{p}^{-1}(q)$. It turns out that this is a metric on $M$, and the topology induced by $d$ coincides with the manifold topology on $M$; in particular $d$ is continuous.

Fix a point $p_{0}\in M$. From now on we will use the notation

$\displaystyle f(p)=d(p,p_{0}).$     (3)

We will say that a horizontal curve $\gamma $ is a geodesic if it is locally length minimizing. By a regular geodesic we will mean a geodesic which is a regular curve.

3. STATEMENT OF THE RESULTS. Let $\varphi :U\longrightarrow \Bbb{ R}$ be a smooth function defined on an open set $U$ in $M$.

DEFINITION 3   The horizontal gradient $\nabla _{H}\varphi $ of $\varphi $ is a smooth horizontal vector field on $U$ such that for each $p\in U$ and $v\in H_{p}$ we have $(\partial _{v}f)(p)=g(v,\nabla _{H}\varphi (p))$.

Using the Schwarz inequality we easily see that the horizontal gradient $%% \nabla _{H}\varphi $ is a direction of the steepest increase of the fuction $%% \varphi $.

The following lemma is well-known and easy in the Riemannian case.

LEMMA 1   Suppose that $f\in C^{\infty }(U)$, where $U$ is an open subset in $M$. Then $\nabla _{H}f$ is a unit vector field on $U$. Moreover, segments in $U$ of minimizing geodesics emanating from $p_{0}$ and parameterized by arc length are exactly the trajectories of $\nabla _{H}f$.

Liu and Sussman in [LS] prove that Hamiltonian geodesics are locally minimizing. Slight modification of their proof by the use of the horizontal gradient leads to a stronger result.

PROPOSITION 1   Let $\gamma $ be a Hamiltonian geodesic. Then each sufficiently short sub-arc of $\gamma $ is the unique length minimizing curve between its endpoints.

The lemma below simply says that if a point $p$ can be joined to $p_{0}$ by a unique minimizing geodesic which is a regular curve, then the sub-Riemannian exponential mapping is ''onto'' a certain neighbourhood of $p$ (the fact being trivial in the Riemannian case).

LEMMA 2   Suppose that $\gamma :[0,L]$ $%% \longrightarrow U$ is a unique minimizing geodesic starting from $%% \gamma (0)=p_{0}$ and parameterized by arc length. Then, if $%% \gamma $ is a regular curve, there exists an open set $U\subseteq M $ such that $\gamma \left( (0,L]\right) \subseteq U$, $%% p_{0}\in \partial U$ and any point from $U$ can be joined to $p_{0}$ by a minimizing regular geodesic.

Now we are prepared to prove the two theorems mentioned in the introduction.

THEOREM 1   Let $\gamma :[0,L]\longrightarrow M$ be a regular geodesic starting from $\gamma (0)=p_{0}$. Then, if $%% L>0$ is sufficiently small, there exists an open $U\subseteq M $ such that $\gamma ((0,L])\subseteq U$, $p_{0}\in \partial U$ and $f$ is smooth on an open and dense subset of $U$.

(Proof. We choose $L>0$ so small that Proposition 1 applies. Next we use lemma 2 and Sard's theorem to the sub-Riemannian exponential mapping to obtain the desired result).

Using the similar methods as in the proof of lemma 2 nad applying lemma 1 we obtain

THEOREM 2   Let $\gamma :[0,L]\longrightarrow M$ be a minimizing geodesic starting from $\gamma (0)=p_{0}$. If $%% f\in C^{\infty }(U)$, where $U$ is such an open set that $%% p_{0}\in \partial U$ and $\gamma ((0,L])\subseteq U$, then $\gamma $ is a regular curve.


Bibliography

Bis
J.-M. Bismut Large Deviations and the Malliavin Calculus, Birkhäuser, Boston, 1984

LS
W. Liu, H.J. Sussman Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distribution, Providence, November 1995

Str
R. Strichartz Sub-Riemannian Geometry, J.Diff. Geom. 24 (1986), pp.221-263

Translated from LaTeX.
2001-05-29