  Botelho, Jamison, and Moln\'ar \cite{BJM}, and Geh\' er and \v{S}emrl
\cite{GeS} have recently described the general form of surjective isometries of
Grassmann spaces of all projections of a fixed finite rank on a Hilbert space
$H$. As a straightforward consequence one can characterize surjective
isometries of Grassmann spaces of projections of a fixed finite corank. In this
paper we solve the remaining structural problem for surjective isometries on
the set $P_\infty (H)$ of all projections of infinite rank and infinite corank
when $H$ is separable. The proof technique is entirely different from the
previous ones and is based on the study of geodesics in the Grassmannian
$P_\infty (H)$. However, the same method gives an alternative proof in the case
of finite rank projections.
