Let $M \subset \mathbb R^n$ be a smooth submanifold. Let us consider $x\in M$ and its tangent space $T_x M$.
We suppose that $T_x M = \mathbb R^n$. Is it true that $M$ contains a neighbourhood of $x$? (neighbourhood in $\mathbb R^n$).
I want to say that it is true but I cannot manage to prove it. I tried the following:
Let suppose that it is not true (but still $T_xM$ is the whole space). Thus, there is $v$ such as $\forall t >0, x+tv \notin M $ (for $t$ small enough). I considered $\gamma : (-\epsilon, \epsilon)\to M$ such that $\gamma(0) = x, \gamma'(0) = v$
We then have $\gamma(t) = x + tv + o(t) \in M$ for $t$ small enough. I want to say that if $t$ is small enough, $o(t)$ doesn't matter and we get a contradiction. But I don't know how to conclude properly.
$\newcommand{\Reals}{\mathbf{R}}$If $T_{x}M = \Reals^{n}$, then the component of $M$ containing $x$ is $n$-dimensional. Consequently, there exist an open neighborhood $V$ of $x$ in $M$, an open set $U$ in $\Reals^{n}$, and a homeomorphism $f:U \to V$. By invariance of domain, $V = f(U)$ is open in $\Reals^{n}$ therefore contains a neighborhood of $x$.
A proof along the proposed lines (filling a neighborhood with curves) looks difficult.