The tangent cone to a set $\mathcal S \subset\Bbb R^n$ at a point $x \in \Bbb R^n$ is the set of all vectors $w \in \Bbb R^n$ for which there exists sequences $x_i \in \mathcal S$ and $\tau_i> 0$, with $x_i\to x$ and $\tau_i\searrow 0 $ such that $w = \lim\limits_{i \to \infty} \frac{x_i - x}{\tau_i}$.
It is well known that if $\mathcal M\subset\Bbb R^,$ is a differentiable embedded submanifold of dimension $k \le n$, then the tangent cone at every point is a $k$-dimensional vector subspace of $\Bbb R^n$ and amounts to the concept of a tangent space.
My question is if $k$-dimensional differentiable embedded submanifolds of $\Bbb R^n$ could be defined as subsets of $\Bbb R^n$ for which the tangent cone at every point is a $k$-dimensional vector subspace of $\Bbb R^n$?
I don't think so The subsets of $\mathbb{R}^n$ with that property seems to include much, much much more than submanifolds.
As a counterexample in $\mathbb{R}^2$, consider a the union of a circle and one of its tangent lines, which intersect at a point $p$.
Since these constructions are local, there is no issue at any point other than $p$. Let $t$ be one of the unit tangents of the circle/line at $p$, and $w\neq 0$ be an element of the tangent cone. There must be sequences $x_i$, $\tau_i$ such that $\lim\frac{x_i-p}{\tau_i}=w$, so eliminating terms as necessary and using the fact that proportional sequences have proportional limits, we also have $w\propto\lim\frac{x_i-p}{\|x_i-p\|}=\pm t$.
As another counterexample which fails to be a submanifold even locally, consider $\mathbb{Q}^m\subset\mathbb{R}^n$.