  The goal of this paper is twofold. First, to give purely local boundary
uniqueness results for maps defined only on one side as germs at a boundary
point and hence not necessarily sending any domain to itself and also under the
weaker assumption that $f(z)=z+o(|z-p|^3)$ holds only for $z$ in a proper cone
in $D$ with vertex $p$. Such results have no analogues in one complex variable
in contrast to the situation when a domain is preserved. And second, to extend
the above results from boundaries of domains to submanifolds of higher
codimension.
