  Let $L^2$ be the Lebesgue space of square-integrable functions on the unit
circle. We show that the injectivity problem for Toeplitz operators is linked
to the existence of geodesics in the Grassmann manifold of $L^2$. We also
investigate this connection in the context of restricted Grassmann manifolds
associated to $p$-Schatten ideals and essentially commuting projections.
