Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on manifolds"),
López de Medrano, S., Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 59. Berlin-Heidelberg-New York: Springer-Verlag, ix, 102 p. (1971). ZBL0214.22501.
but I couldn't find anything on general Grassmannians.