Let F be a field and i < j be integers between 1 and n. A map of
Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is
contained in f(l) for every l in Gr(i, F^n). We show that there are no
continuous nesting maps over C and no algebraic nesting maps over any
algebraically closed field F, except for a few obvious ones. The continuous
case is due to Stong and Grover-Homer-Stong; the algebraic case in
characteristic zero can also be deduced from their results. In this paper we
give new proofs that work in arbitrary characteristic. As a corollary, we give
a description of the algebraic subbundles of the tangent bundle to the
projective space P^n over F. Another application can be found in a recent paper
math.AC/0306126 of George Bergman
