Manifolds with $GL_n(\mathbb{R})$-action.

Question

What is the condition on $n$-dimensional real manifolds in order that they admit an $GL_n(\mathbb{R})$-action in the sense of https://en.wikipedia.org/wiki/Lie_group_action that resembles the canonical $GL_n(\mathbb{R})$-action on $\mathbb{R}^n$ in each coordinate patch, i.e. locally generated by a vector field $V=A_{ij}x_j \frac{\partial}{\partial x_i}$ with $A \in gl_n(\mathbb{R})$? Are there examples where you can "peice together" the canonical action on $\mathbb{R}^n$ to get an action on the total space?

Answer 1

There are several non-trivial examples. For example, the general linear group $GL_n(\mathbb{R})$ acts on the Grassmannians $G_k(\mathbb{R}^n)$ (even transitively). In particular it acts on projective space $\mathbb{P}^{n-1}(\mathbb{R})$. More generally, $GL_n(\mathbb{R})$ acts transitively on flag manifolds.