$M$ is a $k$-dimensional manifold in $\mathbb{R}^n$.
I want to prove that the map $p \mapsto T_pM$ is a smooth map $f:M \to \mathrm{Gr}_{n,k}$.
To prove that is smooth I need to show that for every $p \in M$, there exists a chart $\sigma: U \to \mathrm{Gr}_{n,k}$ around $f(p)=T_pM$ such that $p$ is an interior point of $f^{-1}(\sigma(U)) \subset M$ and such that $\sigma^{-1} \circ f$ is smooth at $p$.
My problem is that I didn't really understand the Grassmannian and I don't know how to find a chart around $T_pM$.
Best Answer
HINT: You can reorder coordinates in $\Bbb R^n$ so that near your point $p$ the manifold $M$ is a graph $y=(x_{k+1},\dots,x_n) = g(x_1,\dots,x_k)$. Then, in these coordinates, you can represent the tangent plane at $x$ as the graph of the derivative $dg_x\colon\Bbb R^k\to\Bbb R^{n-k}$. This is precisely working in one of the Grassmannian charts.
EDIT: Note that the open subset of $G_{n,k}$ consisting of those $k$-planes that project isomorphically onto $\Bbb R^k\times\{0\}\subset\Bbb R^k\times \Bbb R^{n-k} = \Bbb R^n$ is parametrized by the set of all $(n-k)\times k$ matrices $A$. The column vectors of $$\begin{bmatrix} I \\ \hline A \end{bmatrix}$$ give basis vectors for the subspace.