\documentclass[12pt]{amsart}

\usepackage[active]{srcltx}
\usepackage{graphicx}

%%%%%%%% DIMENSIONS %%%%%%%%%%%
\setlength{\textwidth=16 true cm} 
\setlength{\textheight=22.5 true cm}
%\unitlength=1cm
%\topmargin=0pt
\oddsidemargin=0truecm \evensidemargin=0truecm %\mathsurround=2pt
\renewcommand{\baselinestretch}{1.2}
%\frenchspacing
 
%%%%%% FONTS %%%%%%%%%%%%%%
\def\cal#1{\mathcal #1}

%%%%%% Letters %%%%%%%%%%%%%%
%%%%%% BLackboard Letters  %%%%%%%%%%%%
\def\R{{\Bbb R}}
\def\C{{\Bbb C}}
\def\SS{{\Bbb S}}
\def\T{{\Bbb T}}
\def\Z{{\Bbb Z}}
\def\K{{\Bbb K}}
%%%%%% Greek Letters %%%%%%%%%%%%%%
\def\rphi{\varphi}
\def\reps{\varepsilon}
%%%%%% Cal %%%%%%%%%%%%%%
\def\cA{{\cal A}}
\def\cD{{\cal D}}
\def\cF{\cal F}
\def\cG{{\cal G}}
\def\cI{{\cal I}}
\def\cL{{\cal L}}
\def\cM{{\cal M}}
\def\cO{{\cal O}}
\def\cP{{\cal P}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
%%%%%% Gothic letters %%%%%%%%%%%%%%
\def\g{{\mathfrak g}}
\def\gl{{\mathfrak  gl}}
\def\gK{{\mathfrak  K}}
\def\sp{{\mathfrak  sp}}
\def\gU{{\mathfrak  U}}
\def\gV{{\mathfrak  V}}
\def\gX{{\mathfrak  X}}
\def\sp{{\mathfrak  sp}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%% Large fraction %%%%%%%%%%%%%%%%%%%
\def\fracl#1#2{\frac{\displaystyle #1}{\displaystyle #2}}

%%%% Accents %%%%%%%%%%%%
\def\wt#1{\widetilde #1}
\def\wh#1{\widehat #1\,}
\def\ov#1{\overline{#1}}

%%%% Partial derivative, differential operators %%%%%%
\def\pd{\partial}
\def\grad{\mathop{{\rm grad}}}
\def\rot{\mathop{{\rm rot}}}
\def\span{{\rm span}}
%%%%%%%%% Reference to formulas %%%%%%%%%%
\def\fref#1{(\ref{#1})}





%%%%%%%% THEOREMS %%%%%%%%%%%%
\theoremstyle{definition}
\newtheorem{df}{Definition}

\theoremstyle{plain}
\newtheorem{stat}{Statement}
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{lm}{Lemma}
\newtheorem{cor}{Corollary}
\theoremstyle{remark}
\newtheorem{ex}{Example}[section]
%\renewcommand{\theex}{}
\newtheorem{rem}{Remark}


%%%%%%%% COMMENTS (REMOVE THEM IN THE FINAL VERSION!!!) %%%%%%%

\def\comment#1{{\par \tt #1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%% PAPER DATA %%%%%%%%%%%%%%%%%%%
\author{Jos\'e Ricardo Arteaga, Mikhail Malakhaltsev}

\title{Lecture notes on non-holonomic geometry
} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}
\maketitle


\footnotetext{This short course on non-holonomic geometry was delivered at workshop
"IV Taller Geometr\'ia Sub-Riemanniana 
y Operadores de Dirac en espacio-tiempo curvo" (September 24-28, 2007) in Universidad de Los Andes (Bogota, Colombia). 
 }
\section{ Distributions on manifolds}
\label{sec:1}
Let $M$ be an $n$-dimensional smooth manifold. We denote by $\gX(U)$ the Lie algebra of
vector fields defined on an open $U \subset M$.

A \emph{$k$-dimensional distribution} $\Delta$ on $M$ is a smooth field of $k$-dimensional subspaces
$\Delta(p) \subset T_p M$, $p \in M$. The smoothness of this field means that, for 
any $p \in M$, there exists a neighbourhood
$U(p)$ and vector fields $E_1, E_2, \ldots, E_k \in \gX(U(p))$ such that,  
for each $q \in U(p)$, $\Delta(q) = \span\{E_1(q), E_2(q), \ldots, E_k(q)\}$. 

A distribution $\Delta$ on a manifold $M$ is called \emph{totally integrable} if
for each $p \in M$ there exists a submanifold $\Sigma$ passing through $p$ such that 
$T_q \Sigma = \Delta(q)$, for any $q \in \Sigma$. This submanifold is called an \emph{integral 
submanifold} of $\Delta$.
A totally integrable distibution  is called  a foliation, 
and the integral submanifolds are called the leaves.

Locally, a distribution $\Delta$ can be given either by vector fields $E_1, \ldots, E_k$
spanning the subspaces $\Delta(p) \subset T_p M$, or by differential forms 
$\omega^1, \omega^2, \ldots, \omega^{n-k}$ such that, 
for any $V \in T_p M$, $V \in \Delta(p)$ if and only if $\omega^\alpha(V)=0$, 
$\alpha=1,\ldots,n-k$. 

The following theorem gives necessary and sufficient conditions for a distribution 
be completely integrable.

\begin{thm}[Frobenius]
\label{thm:3}
Let $\Delta$ be a distribution on a manifold $M$ locally 
given by vector fields $E_a$, $a=1, \ldots, k$, or by fields of 
1-forms $\omega^\alpha$,
$\alpha=1,\ldots,n-k$. 
The following conditions are equivalent:

\emph{a)} The distribution $\Delta$ is completely integrable\emph{;}

\emph{b)} $[E_a, E_b] = Q^c_{ab} E_c$, where $Q^c_{ab}$ are functions\emph{;}

\emph{c)} $d\omega^\gamma =   Q^\gamma_\beta \wedge\omega^\beta$, where  $Q^\gamma_\beta$
are 1-forms. 
\end{thm}

\begin{ex}
Consider a distribution $\Delta$ on $\R^3$  given by  $\omega = dz - xdx - ydy$.
Then $d\omega = 0$, and, by Theorem \ref{thm:3}, $\Delta$ is completely integrable.
In fact,  this distribution 
consists of planes tangent to hyperbolic paraboloids 
$z = \frac{1}{2}(x^2 + y^2)+c$. 
\end{ex}
\begin{ex}
Let a distribution $\Delta$ on $\R^3$ be given by $\omega = dy - zdx$.
Then the vector fields  
$E_1 = \frac{\pd}{\pd x}+z\frac{\pd}{\pd y}$, $E_2 = \frac{\pd}{\pd z}$
span $\Delta$.
We have $[E_1,E_2] = -\frac{\pd}{\pd y}$, and this vector field does not lie in $\Delta$ because
$\omega(\frac{\pd}{\pd y}) = -1 \ne 0$. Hence this distribution is not completely integrable.
This means that one cannot find a surface tangent to $\Delta$.
\end{ex}

\section{Non-holonomic surface in $E^3$}
\label{sec:2}

Let $E_3$ be the three-dimensional Euclidean space, we denote by $(\cdot, \cdot)$ the scalar product in $E_3$. 
Any  distribution $\Delta$ on $E_3$ can be given by a unit vector field $\vec n$ normal to $\Delta$.
The integrability condition for $\Delta$ can be formulated in terms of $\vec n$.
\begin{thm}
$\Delta$ is integrable if and only if $(\vec n, \rot\vec n) = 0$.
\end{thm}

\subsection{Geometric sense of non-holonomity}
The non-holonomity of a distribution in $E^3$ 
can be visualized in the following manner. 
Take a small disk $L$  in the distribution plane and draw the straight line $l(A)$ through
any point $A$ of the disk boundary $\pd L$ in the direction of $\vec n(A)$. Thus we obtain a
ruled surface. Now take a curve $\gamma$ passing through $A_0$ which is
orthogonal to the straight lines $l(A)$ (see Fig.~1). 
\begin{figure}[h]
\includegraphics[scale=0.4]{1.eps}
\caption{Geometric sense of non-holonomity. I}
\end{figure}
This curve again meets $l(A_0)$ at a point $A_1$, then 
 $\frac{|A_0 A_1|}{\sigma(L)}$,  where $\sigma(L)$ is the area of $L$, converges to 
$(\vec n, \rot \vec n)_{A_0}$  as $L$ contracts to $A_0$. 
If $\Delta$ is holonomic (totally integrable), then $\gamma$ lies on the integral surface passing 
through $A_0$, hence $A_1 = A_0$, and $|A_0 A_1|=0$.

Another way to visualize the non-holonomity is to take two unit orthogonal vector
fields $\vec a$, $\vec b$ spanning $\Delta$.
Let $\gamma(t)$ be an integral curve of the field $\vec b$, and for each $t$ we take 
the integral curve  $\delta_t(s)$ of the field $\vec a$ passing through $\gamma(t)$. Thus we obtain a
surface $\Sigma$ (see Fig.~2). Now at a point $A(s)=\delta_{t(s)}(s)$ we take the vector $\vec\nu(A(s))$ 
normal to $T_{A(s)}\Sigma$  and the vector $\vec n(A(s))$ normal to $\Delta(A(s))$. 
Denote by $\varphi(s)$ the angle between $\vec\nu(A(s))$ and $\vec n(A(s))$.
Then 
\begin{equation}
\frac{d\varphi}{ds} = - (\vec n, \rot \vec n).
\label{eq:18}
\end{equation}
If $\Delta$ is holonomic (totally integrable), then $\Sigma$ is an integral manifold 
of $\Delta$, and $T_p\Sigma=\Delta(p)$.  Hence $\vec\nu(p)=\vec n(p)$, and $\varphi=0$.


\begin{figure}[h]
\includegraphics[scale=0.4]{2.eps}
\caption{Geometric sense of non-holonomity. II}
\end{figure}


\subsection{Curvatures of non-holonomic surface}

\subsubsection{Curvatures of holonomic surface}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{3.eps}
\end{center}
\caption{Moving frame}
\label{fig:3}
\end{figure}
Let a surface $\Sigma$ in $E_3$ be given by the parametric equation 
$\vec r = \vec r(u^1,u^2)$. 
Let us set $\vec r_i = \pd \vec r_i = \frac{\pd}{\pd u^i}\vec r$.
The vectors  $\vec r_1$, $\vec r_2$ give a frame of the tangent plane $T\Sigma$, and the 
coordinates of the metric tensor (the first fundamental form) with respect to this frame are
 $g_{ij} = (\vec r_i, \vec r_j)$. We denote by $g^{ij}$ the tensor inverse to $g_{ij}$,
this means that $g^{is} g_{js} = \delta^i_j$.
Now   let $\vec n$ be the unit normal to $T\Sigma$.
Then we obtain a moving frame $\{\vec r_1, \vec r_2, \vec n\}$ (see Fig. \ref{fig:3}) and the derivation equations:
\begin{eqnarray}
\label{eq:1}
\pd_i \vec r_j = \Gamma^k_{ij} \vec r_k + h_{ij}\vec n
\\
\label{eq:2}
\pd_i \vec n = -h_i^j \vec r_j
\end{eqnarray}
Here 
\begin{equation}
\Gamma^k_{ij} = \frac{1}{2}g^{ks}(\pd_i g_{js} + \pd_j g_{is} - \pd_s g_{ij})
\label{eq:3}
\end{equation}
are the \emph{connection coefficients} (we will discuss them later), which depend only on 
the metric tensor, $h_{ij}$ is the \emph{second fundamental form}, and $h^i_j = g^{is} h_{js}$ 
is the \emph{shape operator}. 

The shape operator is a symmetric linear operator on $T\Sigma$, therefore it has real 
eigenvalues $k_1$, $k_2$ which are called \emph{principal curvatures} of $\Sigma$. 
$H = \frac{1}{2}(k_1 + k_2)$ is called the \emph{mean curvature} of $\Sigma$ and $K=k_1 k_2$ the 
\emph{total (Gaussian) curvature} of $\Sigma$. 

Geometrically, the principal curvatures can be described as follows.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{4.eps}
\end{center}
\caption{Normal curvature}
\label{fig:4}
\end{figure}
Take a point $p \in \Sigma$. For any direction tangent to $\Sigma$ given by a unit vector $\vec\tau$ 
at $p$, we 
can take the plane passing through $\vec n$ and $\vec \tau$ (a normal plane), 
and this plane meets the surface $\Sigma$ along a normal section $\gamma(\vec\tau)$.
Let  $\vec \rho(s)$ be the parametric equation of $\gamma(\vec\tau)$ referred to the 
natural parameter $s$. Then, $k_n(\vec\tau) = (\frac{d^2}{ds^2}\vec \rho(s), \vec n)$
is called the \emph{normal curvature of $\Sigma$ with respect to the direction $\vec\tau$}.
As $k_n$ is a continuous function on the circle, it attains its extremal values. One 
can easily prove that the principal curvatures $k_1$ and $k_2$ are just the extremal 
values of the normal curvature $k_n(\vec \tau)$. In addition, the normal curvature can be written 
in terms of the second fundamental form: $k_n(\vec\tau) = h(\vec\tau,\vec\tau)$.
Also, using \fref{eq:2}, we get 
\begin{equation}
k_n(\vec\tau) = - (\tau^i \pd_i \vec n, \vec \tau) =-(\nabla_{\vec\tau}\vec n, \vec \tau). 
\label{eq:4}
\end{equation}
where $\nabla$ stands for the directional derivative.

The total curvature can be defined in another way. The unit normal
$\vec n$ determines the Gaussian map 
$G : \Sigma \to \SS^2$, $p \in \Sigma \mapsto \vec n(p)$.
Let $U$ be a neighborhood of $p \in \Sigma$, and $V=G(U)$ be its image on  
$\SS^2$. We denote by $\sigma(U)$ the area of $U$ on $\Sigma$, and by $S(V)$ the area of $V$
on $\SS^2$.
Then the total curvature $K$ at $p$ equals  $\lim \frac{S(V)}{\sigma(U)}$ as 
$U$ contracts to $p$. From this property one can obtain another formula for 
the total curvature:
\begin{equation}
K = \frac{(\pd_1 \vec n, \pd_2 \vec n, \vec n)}{\sqrt{g_{11}g_{22}-g_{12}^2}}
 \label{eq:5}
\end{equation}
If we have a foliation in $E^3$ whose leaves are  given by an implicit 
equation $\Phi(x^1, x^2,x^3)=const$, then the vector field $\vec n$ is defined
in the entire $E^3$. In this case, from \fref{eq:5} it follows that 
\begin{equation}
K = (\vec P, \vec n), 
\label{eq:6}
\end{equation}
where $\vec P$ is  the \emph{curvature vector} defined as follows:
\begin{equation}
\vec P = \big(
(\frac{\pd}{\pd x^2} \vec n, \frac{\pd}{\pd x^3} \vec n, \vec n),
(\frac{\pd}{\pd x^3} \vec n, \frac{\pd}{\pd x^1} \vec n, \vec n),
(\frac{\pd}{\pd x^1} \vec n, \frac{\pd}{\pd x^2} \vec n, \vec n)
\big),
\label{eq:7}
\end{equation}
where $(\cdot,\cdot,\cdot)$ denotes the mixed product of three vectors.
In particular, \fref{eq:6} implies the von Neumann formula:
\begin{equation*}
K = 
\frac{
\left|
\begin{array}{cccc}
\pd_{11}\Phi & \pd_{12}\Phi & \pd_{13}\Phi & \pd_{1}\Phi 
\\
\pd_{21}\Phi & \pd_{22}\Phi & \pd_{23}\Phi & \pd_{2}\Phi 
\\
\pd_{31}\Phi & \pd_{32}\Phi & \pd_{33}\Phi & \pd_{3}\Phi 
\\
\pd_{1}\Phi  & \pd_{2}\Phi  & \pd_{3}\Phi  &      0 
\end{array}
\right|
}{
(\pd_1\Phi)^2 + (\pd_2\Phi)^2 + (\pd_3\Phi)^2)^2
},
\end{equation*}
where the partial derivatives are taken with respect to $x^1$, $x^2$, $x^3$. 

\subsubsection{Curvatures of non-holonomic surface. First type.}
Let a two-dimensional distribution (a non-holonomic surface) $\Delta$ 
be given in $E^3$, and $\vec n$ be the unit normal vector field of $\Delta$. 
The formula \fref{eq:4} gives us the normal curvature of a surface in terms of 
the normal vector $\vec n$ and a tangent vector $\vec\tau$, so it can be generalized
to the non-holonomic case. We say that the \emph{normal curvature of non-holonomic surface
$\Delta$ in the direction of a unit vector $\tau \in \Delta$ } is 
\begin{equation}
k_n(\tau)  =-(\nabla_{\vec\tau}\vec n, \vec \tau). 
\label{eq:8}
\end{equation}
Let $A \in E^3$, and $\vec e_1$, $\vec e_2$ be an orthonormal frame of $\Delta(A)$. Then
$\vec \tau(\varphi) = \cos \varphi \vec e_1 + \sin \varphi \vec e_2$,
and one can easily see that $k_n(\vec \tau)$ defined by \fref{eq:8} has the form
\begin{equation}
k_n(\vec\tau(\varphi)) = k_1 \cos\varphi + k_2 \sin\varphi,
\label{eq:9}
\end{equation}
and it is clear that $k_1$ and $k_2$ are extremal values of $k_n(\vec\tau(\varphi))$.
$k_1$ and $k_2$ are called \emph{the principal curvatures of $\Delta$ of first type}. 
Then we can define the \emph{mean curvature of first type} 
$H_{I} = \frac{1}{2}(k_1+k_2)$ and the \emph{total curvature of first type} 
$K_{I} = k_1 k_2$.


\subsubsection{Curvatures of non-holonomic surface. Second type.}
For a non-holonomic distribution $\Delta$ we can 
define the \emph{total curvature of second type} using \fref{eq:6}, where $\vec P$ is given by \fref{eq:7}.
We will denote it by $K_{II}$.

At the same time, this curvature can be obtained in the following way. Let us consider eigenvalues of
the Jacobi matrix $J(\vec n)=||\frac{\pd n^i}{\pd x^j}||$, 
where $n^i$ are the Cartesian coordinates of the vector-function $\vec n$. 
Since $\vec n$ maps the three-dimensional space into the two-dimensional sphere, the determinant of  $J(\vec n)$
is zero, hence one of the eigenvalues of $J(\vec n)$ is zero. We will call the other two eigenvalues 
$\lambda_1$, $\lambda_2$ the principal curvatures of second type (they can be complex conjugate) and one can prove that
$K_{II} = \lambda_1 \lambda_2$. Also, we can define the \emph{mean curvature of second type} to be
$H_{II} = \frac{1}{2}(\lambda_1+\lambda_2)$. 
\begin{thm}
For a non-holonomic surface $\Delta$ we have
\begin{equation}
H_{I} = H_{II} \qquad K_I - K_{II} = \frac{(\vec n, \rot \vec n)^2}{4}.
\label{eq:10}
\end{equation}
\end{thm} 

\begin{ex}
An example of distribution which has $K_{II}=0$, $H_{II}=0$, and is non-holonomic can 
be constructed as follows. We take the family of concentric spheres in $E^3$ with center 
at the origin, and each sphere $\SS^2(r)$ of the family is uniquely determined by its radius $r>0$.
For each $r>0$, we take a straight line $l(r)$ lying in the plane $XOY$ such that the angle between 
$l(r)$ and $OX$ is $\alpha(r)$, where $\alpha(r)$ is a function (see Fig. \ref{fig:6}).  
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{6.eps}
\end{center}
\caption{non-holonomic distribution with zero curvatures}
\label{fig:6}
\end{figure}
Now, for each  $\SS^2(r)$, we take the unit vector field $\vec n$ tangent to this sphere which is 
generated by rotations about the axis $l(r)$ (certainly this field is not defined 
in the points where $l(r)$ meets $\SS^2(r)$). Thus we obtain a unit vector field $\vec n$ 
which is defined everywhere in $E^3$ except for points where  $l(r)$ meets $\SS^2(r)$.
If $\frac{d\alpha}{dr} \ne 0$, at points with $z \ne 0$ we have 
$(\vec n, \rot \vec n) \ne 0$, so the distribution $\Delta$ orthogonal to $\vec n$ is 
non-holonomic. At the same time, the second type mean and total curvature of $\Delta$ vanish.
\end{ex}

\begin{ex}
Another way to obtain a distribution $\Delta$ with zero total curvature of second type
is to consider a surface $\Sigma$ with a vector field $\vec a$ along it (we  assume
that $\vec a$ is not tangent to $\Sigma$). Then, for each point $p \in \Sigma$ we take 
a plane curve which is tangent to $\vec a(p)$ at $p$. Thus, a neighborhood of $\Sigma$  
is foliated by these curves, and for the vector field $\vec n$ we take the unit  vector 
field tangent to these curves. The distribution $\Delta$ orthogonal to $\vec n$ has $K_{II}=0$.
\end{ex}

\subsubsection{Gauss-Bonnet formula}
For a closed surface $\Sigma$ in $E^3$, we have the Gauss-Bonnet formula
\begin{equation}
\int_\Sigma K dS = 2 \pi \chi(\Sigma), 
\label{eq:11}
\end{equation}
where $K$ is the total curvature of $\Sigma$, and  $\chi(\Sigma)$ is the Euler characteristic 
of $\Sigma$.

For a non-holonomic surface $\Delta$, we have the following generalization of the Gauss-Bonnet 
formula. 
\begin{thm}
\label{thm:1}
Let $Q$ be a closed oriented surface which lies in the domain of $\Delta$. Then 
\begin{equation}
\int_Q (K_{II} \vec n + 2 H_{II} \vec k + \nabla_{\vec k} \vec n, \vec\nu) dS = 
4\pi \theta,
\label{eq:12}
\end{equation}
where $\vec k$ is the curvature vector of integral curves of $\vec n$, $\vec \nu$ is the unit normal
to the surface $Q$, $\nabla_{\vec k}$ is the derivative in $E^3$ with respect to $\vec k$, 
and $\theta$ is the degree of the map $\vec n |_Q : Q \to \SS^2$. 
\end{thm}
Let us recall that the degree of a map $f : Q \to P$ 
between oriented surfaces is an integer which equals zero if $f$ is homotopic to 
a constant map. Under assumptions of Theorem \ref{thm:1}, $Q$ is a boundary of a region
$V \subset E^3$. If $\Delta$ is defined everywhere in $V$, and $V$ is contractible, 
then $\vec n$ is homotopic to a constant vector field, hence $\theta=0$.


 \subsection{The shortest and the straightest curves on a non-holonomic surface}

Let us recall that a geodesic on a surface $\Sigma$ in the three-dimensional Euclidean space
can be defined in two ways: a) a geodesic is a straightest curve; b) a geodesic is a locally
shortest curve. Consider these definitions in more details.

 Let us consider a curve  on a surface $\Sigma$ given by a parametric equation 
$\vec r = \vec \rho(s)$, where $s$ is the natural parameter. The length of projection 
of the curvature vector $\frac{d^2}{ds^2}\vec\rho$ onto the plane tangent to $\Sigma$ 
is called the \emph{geodesic curvature}. If the geodesic curvature is zero, or equivalently 
the curvature vector is collinear to the normal vector of the surface, we say that this 
curve is a \emph{straightest curve} on $\Sigma$ because it has no ``acceleration'' along the surface.

At the same time, if a curve $\gamma$ joining sufficiently close points $p$ and $q$ on the 
surface $\Sigma$ has the minimal length among all the curves joining these points, then 
$\gamma$ is called the \emph{shortest curve}.

For a surface $\Sigma \subset E^3$,  the definition a) is equivalent to definition b). However, for 
a non-holonomic surface $\Delta$ the curves defined by a) and b) are different.
We will call a curve $\gamma$ in $E^3$ an \emph{admissible curve} if its tangent vector $\frac{d}{dt}\gamma$
lies in $\Delta(\gamma(t))$. 

\subsubsection{Straightest lines on non-holonomic surfaces}
Let a non-holonomic surface $\Delta$ be given. 
An admissible curve $\gamma$ is called a \emph{straightest curve} if its curvature vector is collinear to 
the normal vector $\vec n$ of $\Delta$. 

\begin{thm}
Let $\vec r = \vec \rho(s)$ be parametric equation of an admissible curve $\gamma$.
Then $\gamma$ is a straightest line if and only if $\vec \rho(s)$ satisfies 
	the following second order  differential equation:
\begin{equation}
\frac{d^2\vec\rho}{ds^2}  = 
- \left(\frac{d\vec\rho}{ds}, \nabla_{\frac{d\vec\rho}{ds}}  \vec n\right) \vec n.
\label{eq:13}
\end{equation}
\end{thm}
Thus, given a point $p \in E^3$ and a vector $V_0 \in \Delta(p)$ one 
can find a unique straightest curve $\gamma$ which passes through $p$ 
and whose tangent vector at $p$ is $V_0$.  

\subsection{Shortest lines on non-holonomic surfaces}

Let us consider two points $p$ and $q$ in $E^3$, and let $P_a(p,q)$ be the set of
all admissible curves joining $p$ and $q$. We say that a curve $\gamma \in P_a(p,q)$ 
such that $\gamma(a)=p$,  $\gamma(b)=q$,
is a shortest admissible curve if the length $L(\gamma) = \int_a^b |\frac{d\gamma}{dt}| dt$
is less than or equal to the length of any other curve in $P_a(p,q)$.
\begin{thm}
Let a curve $\gamma \in P_a(p,q)$ be given by a parametric equation $\vec r = \vec \rho(s)$,
$s \in [a,b]$.
Then, if $\gamma$ is a  shortest curve, then $\vec \rho(s)$
satisfies the third order ordinary differential equation 
\begin{equation}
\frac{d^2\vec \rho}{ds^2} + 
\frac{d}{ds}
\left\{
\frac{(\frac{d^2\vec \rho}{ds^2}, \frac{d\vec \rho}{ds}, \vec n)}{(\vec n, \rot \vec n)}
\right\} \vec n +
\frac{(\frac{d^2\vec \rho}{ds^2}, \frac{d\vec \rho}{ds}, \vec n)}{(\vec n, \rot \vec n)}
[\rot \vec n, \frac{d\vec \rho}{ds}] = 0.
\label{eq:14}
\end{equation}
\end{thm}
Therefore, given a point $p \in E^3M$ and a vector $V_0 \in \Delta(p)$ one can find 
an infinite number of shortest curves joining $p$ and $q$  which pass through $p$ 
and whose tangent vector at $p$ is $V_0$.  

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{5.eps}
\end{center}
\caption{A unique straightest curve and a bundle of shortest curves}
\label{fig:5}
\end{figure}


\section{Properties of metric of sub-Riemannian manifold}
\label{sec:3}

\subsection{Accessibility of points}
Let $M$ be an $n$-dimensional  differential manifold, and $\Delta$ be an $m$-dimensional 
distribution on $M$. We will call a curve $\gamma$ in $M$ \emph{admissible} if its tangent vector
lies in $\Delta$. Let us denote by $P_a(p,q)$ the set of all admissible curves joining 
$p$ and $q$ (this set can be empty!).  
We say that a point $q$ is \emph{accessible} from a point $p$ if  $P_a(p,q)$ is not empty. 

\begin{thm}[Sussman]
The set of points accessible from a given point $p$ is an immersed submanifold in $M$.
\end{thm}

\subsection{Completely non-integrable distributions}
First let us define a  distribution with singularities on an $n$-dimensional manifold
$M$.  A \emph{distribution with singularities} $\Delta$ on $M$ is a smooth field of subspaces
$\Delta(p) \subset T_p M$, $p \in M$. The smoothness of this field means that, for 
any $p \in M$, there exists a neighbourhood
$U(p)$ and vector fields $E_1, E_2, \ldots, E_k \in \gX(U(p))$ such that,  
for each $q \in U(p)$, $\Delta(q) = \span\{E_1(q), E_2(q), \ldots, E_k(q)\}$. 
Note that we do not assume that the vectors $E_1(q), E_2(q), \ldots, E_k(q)$ are linearly independent 
(this makes difference from the definition of $k$-dimensional distribution, see Section \ref{sec:1}).
Certainly, each $k$-dimensional distribution is a distribution with singularities.

 

Now let $\Delta$ be a distribution  with singularities on $M$. 
Let $\gX_\Delta(U)$ be the set of vector fields $X$ on an open set $U \subset M$ such
that, for each $q \in U$, $X(q) \in \Delta(q)$.
Now,  for the distribution $\Delta$ we can construct another distribution $\Delta^1$ with 
singularities  in the following way. For any point $p \in M$, we set
\begin{equation}
\Delta^1(p) = \{X(p), [X,Y](p) \mid X,Y \in \gX_\Delta(U_p), \text{ where } U_p 
\text{ is a neighborhood of } p\}
\label{eq:15}
\end{equation}
From this definition it follows that $\Delta(p) \subset \Delta^1(p)$, for any 
$p \in M$. 

It is clear that, if $\Delta$ is completely integrable if and only if $\Delta^1=\Delta$.

For a distribution $\Delta$ we can sequentially construct the series $\Delta^k$ of 
distributions with singularities by setting $\Delta^{k+1} = (\Delta^k)^1$. Since
$\Delta^k \subset \Delta^{k+1}$, and $\Delta^k = \Delta^{k+1}$ if and only if 
$\Delta^k$ is completely integrable, there exists $N$ such that 
$\Delta^N = \Delta^{N+1} =  \Delta^{N+2} = \ldots$, and $\Delta^N$ is completely integrable.
Thus, if the subspaces $\Delta^N(p)$ do not coincide with $T_p M$, then we get a foliation 
(with singularities) on $M$. If, for each $p \in M$, we have $\Delta^N(p) = T_p M$, then we say
that the distribution $\Delta$ is \emph{completely non-integrable}.

If a distribution $\Delta$ is completely non-integrable, then the following theorem 
implies that any point of $M$ is accessible from any other point of $M$. 

\begin{thm}[Caratheodory, Rashevskii, Chow]
\label{thm:2}
Let a distribution $\Delta$ on a connected manifold $M$ be completely non-integrable.
Then, for any $p$, $q$ in $M$, $P_a(p,q) \ne \emptyset$. 
\end{thm}

\subsection{Sub-Riemannian metric of completely non-integrable distributions}

A \emph{sub-Riemannian manifold} is a completely non-integrable distribution $\Delta$ 
endowed by a metric $g$. This means that, for any $p$ in $M$, on $\Delta(p)$ 
we have the scalar product $g_p : \Delta(p) \times \Delta(p) \to \R$ such that 
for any $X, Y \in \gX_\Delta(M)$ the function $p \mapsto g_p(X(p), Y(p))$ is smooth. 
In other words, we can say that we have a metric on $\Delta$ considered as a vector subbundle
of the vector bundle $TM$.

If $\gamma(s)$, $s \in [a,b]$, is an admissible curve in $M$, then we can define the length of 
$\gamma(s)$ as follows:
\begin{equation}
L(\gamma) = \int_{a}^{b} g_{\gamma(s)}(\frac{d\gamma}{ds},\frac{d\gamma}{ds}) ds
\label{eq:16}
\end{equation}


From  theorem \ref{thm:2} it follows that we can define a distance $d$ on $M$:
\begin{equation}
d(p,q) = \inf_{\gamma \in P_a(p,q)} L(\gamma).
\label{eq:17}
\end{equation}

This distance has some nice properties like the distance defined by a Riemannian metric.
For example, we have the following theorem.
\begin{thm}
The topology defined by the distance $d$ coincides with the original topology on $M$.
\end{thm}

A curve $\gamma(s)$, $s \in [a,b]$, is called a minimizing geodesic if
$L(\gamma)= d(\gamma(a), \gamma(b))$. 

\begin{thm}[Hopf-Rinow theorem for sub-Riemannian manifolds]
\emph{a)} Sufficiently near points can be joined by a minimizing geodesic;

\emph{b)} If $(M,d)$ is a complete metric space, then any two points can be joined by a minimizing 
geodesic.
\end{thm}

\section{References}

The detailed exposition of results on geometry of non-holonomic surfaces in the three-dimensional
Euclidean space (stuff of section \ref{sec:2}) 
can be found in the very interesting and profound book by Yu. Aminov:

Yu. Aminov:
\emph{The geometry of vector fields.}  Amsterdam: Gordon and Breach Publishers. (2000). 

For the introduction to the sub-Riemannian geometry (stuff of section \ref{sec:3}), 
we can recommend the book:

R. Montgomery \emph{A Tour of Subriemannian geometries, Their Geodesics and Applications },
Mathematical Surveys and Monographs, Vol. 91, AMS, 2002.


\end{document}

