Shubert cells & the flag manifold The geometry of the flag manifold is a deep and beautiful topic, which should be broached at an early stage. the real story is: The natural metric on the flag manifold yields a Morse function (for any base point). The cells for this are the Shubert cells. This function (and cells) descend to Grassmannians too, including projective space. Remember, it's always comparing your subspace -with a given complete flag-. (it was much more ad hoc when I heard it described) Indeed, the standard minimal CW cx structure on RP^n and CP^n come from Shubert cells (i.e., this Morse function); also the minimal CW cx structure on S^n comes from a Morse function (but that's a different one: it's distance, or equivalently just height under the standard embedding)