  We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and
${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex,
respectively real, analytic curve in a discrete set. The examples are realized
either as compact tori or as properly imbedded Euclidean spaces, and are the
graphs of quasianalytic functions. In the complex case, these submanifolds
contain real $n$-dimensional tori or Euclidean spaces that are not pluripolar
while the intersection with any complex analytic disk is polar.
