  Let us say that a curve $C\subset\mathbb P^3$ is osculating self-dual if it
is projectively equivalent to the curve in the dual space $(\mathbb P^3)^*$
whose points are osculating planes to~$C$. Similarly, we say that a
$k$-dimensional subvariety $X\subset\mathbb P^{2k+1}$ is osculating self-dual
if its second osculating space at the general point is a hyperplane and $X$ is
projectively equivalent to the variety in $(\mathbb P^{2k+1})^*$ whose points
are second osculating spaces to $X$.
  In this note we show that for each $k\ge 1$ there exist many osculating
self-dual $k$-dimensional subvarieties in $\mathbb P^{2k+1}$.
