For any k-dim. subspace $ L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Question

Let $M$ be an $n$ dimensional manifold, and $S\subseteq M$ be a k-dim. sub manifold of $M$, where each is in fact a smooth manifold to be precise.

We know that $T_p S$ is a k-dim. subspace of $T_p M$. Let us fix an atlas for $M$, then...

Question:

For any k-dim. subspace $ L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Answer 1

Yes, we can. Let $\varphi\colon\mathbb R^n\longrightarrow M$ be a chart of the manifold $M$ such that $\varphi(0)=p$. Then $D\varphi(0)^{-1}(L)$ is a $k$-dimensional subspace of the tangent space of $\mathbb R^n$ at $0$ (which is $\mathbb R^n$, of course). And in $\mathbb R^n$ it is clear that you can find a submanifold $N$ such that $0\in N$ and that its tangent space at $0$ is $D\varphi(0)^{-1}(L)$. So, take $R=\varphi(N)$.