Flag manifold is manifold

Question

I'm looking for the easiest way to see/prove that flag manifold is indeed a manifold.

Any help is welcome. Thanks in advance.

Answer 1

As suggested in the comments, one systematic way to proceed is to observe that the general linear group $G=\mathrm{GL}_n(\mathbf{R})$ acts transitively on the set of complete flags in $\mathbf{R}^n$, and the stabilizer of the most obvious flag is the closed subgroup $B$ consisting of invertible upper triangular matrices. Thus there is a bijection (of sets) between $G/B$ and the set of complete flags.

Now it is a general fact that for a Lie group $G$ and a closed subgroup $B$, the quotient $G/B$ carries a unique manifold structure such that the projection is a $C^\infty$ map admitting local smooth sections (see e.g. Theorem 3.58 of Warner's book Foundations of differentiable manifolds and Lie groups). Thus while you might wonder if there are other manifold structures on the set of complete flags, there is a unique one that is compatible with the action of the general linear group in the sense just mentioned.