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\section*{ The problems to start with}

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1. (B. Uribe) 
{ \bf Is it true that a straightest line on a non-holonomic surface is 
a shortest line?}

On a two-dimensional surface $\Sigma$ the geodesics can be characterized in 
the following way. 

a) A geodesic is a straightest line. This means that if $\vec\rho=\vec\rho(s)$ is
the equation of a geodesic on $\Sigma$ 
given with respect to the natural parameter (the 
arc length), then  $\vec\rho''(s)$ is collinear to the normal vector of $\Sigma$.

b)  A geodesic is a shortest line. This means that, for any two sufficiently
closed points, the geodesic is the curve of minimal length among all the curves
joining these points.  
 
For a holonomic surface a) is equivalent to b). However, for a non-holonomic
surface a) gives an ODE of second order for $\rho(s)$ and b) g gives an ODE 
of third order for $\rho(s)$. These equations look different, however it is 
possible that any solution of the second order equation is a solution to the
third order equation, this would mean that {\bf a straightest line is a shortest line}.
 
At the same time, even if this is not true in general, one can find classes 
of non-holonomic surfaces for which a) is equivalent b), or implies b). 


2.  {\bf Can  the total curvature be obtained from the Riemannian curvature tensor?}

It is known that the total curvature of a surface is expressed in terms of 
the curvature tensor of the Riemannian connection induced on the surface,
namely it coinsides with the scalar curvature. For a non-holonomic surface,
we also have the total curvature (of first and second type), the 
partial linear connection induced on the surface, and the prolongation 
of this connection.
We can construct the curvature tensor of this connection and of its prolongation.
The question is: can  the total curvature (of first or second type) be 
obtained from these curvature tensors? 
 
3. {\bf Flows of non-holonomic surfaces}

A flow of non-holonomic surfaces in $E_3$ is a family $\Delta_t$ 
of 2-dimensional distributions with induced metrics $g_t$. One can 
consider a ''Ricci flow'' of non-holonomic surfaces, i.\,e.
$\frac{d}{dt} g = K g$, where $K$ is the curvature of first or second 
type. One can ask if the Ricci flow $\Delta_t$ converges to a non-holonomic
surface $\Delta$, and, if it converges, what are the properties  
of $\Delta$ and of the limiting metric on $\Delta$, in particular, can $\Delta$ be holonomic?

Another question is if one can find natural geometric conditions on a flow 
$\Delta_t$ of non-holonomic surfaces under which this flow converges 
to a holonomic surface. 

And, finally, what is a Ricci flow of sub-Riemannian manifolds?
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