  Narasimhan and Ramadas showed that the restricted holonomy group of the
Coulomb connection is dense in the connected component of the identity of the
gauge group when one considers the product principal bundle $S^3 \times
SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold
of dimension $n\ge 2$ with a boundary and use Dirichlet boundary conditions on
connections as defined by Marini. A key step in the method of Narasimhan and
Ramadas consisted in showing that the linear space spanned by the curvature
form at one specially chosen connection is dense in the holonomy Lie algebra
with respect to an appropriate Sobolev norm.
  Our objective is to explore the effect of the presence of a boundary on this
construction of the holonomy Lie algebra. Fixing appropriate Sobolev norms, it
will be shown that the space spanned, linearly, by the curvature form at any
one connection is never dense in the holonomy Lie algebra. In contrast, the
linear space spanned by the curvature form and its first commutators at the
flat connection is dense and, in the $C^\infty$ category, is in fact the entire
holonomy Lie algebra. The former, negative, theorem is proven for a general
principle bundle over $M$, while the latter, positive, theorem is proven only
for a product bundle over the closure of a bounded open subset of
$\mathbb{R}^n$. Our technique for proving absence of density consists in
showing that the linear space spanned by the curvature form at one point is
contained in the kernel of a linear map consisting of a third order
differential operator, followed by a restriction operation at the boundary;
this mapping is determined by the mean curvature of the boundary.
