  Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the
Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim
H>2k$ and $f$ is an orthogonality preserving injective transformation of
${\mathcal G}_{k}(H)$, i.e. for any orthogonal $X,Y\in {\mathcal G}_{k}(H)$ the
images $f(X),f(Y)$ are orthogonal. If $\dim H=n$ is finite, then $n=mk+i$ for
some integers $m\ge 2$ and $i\in \{0,1,\dots,k-1\}$ (for $i=0$ we have $m\ge
3$). We show that $f$ is a bijection induced by a unitary or anti-unitary
operator if $i\in \{0,1,2,3\}$ or $m\ge i+1\ge 5$; in particular, the statement
holds for $k\in \{1,2,3,4\}$ and, if $k\ge 5$, then there are precisely
$(k-4)(k-3)/2$ values of $n$ such that the above condition is not satisfied. As
an application, we obtain a result concerning the case when $H$ is
infinite-dimensional.
