Given the parametrization of a smooth $m$-dimensional manifold $M$ in $\mathbb{R}^n$, $n\geq m$, how can one find the minimal dimension $p\geq n$ that $M$ really "explores" in $\mathbb{R}^n$ (that is, what is the smallest $p$ such that $M$ is in $\mathbb{R}^p$).
I think my main problem is that I cannot find the correct terminology for my question, which might be unclear, so here are two examples. Let $M_1 \subset \mathbb{R}^3$ the curve parametrized by $(\cos(t),\cos(t),\sin(t))$, $t\in[0,2\pi[$. This is clearly an ellipse, contained in the plane defined by $e_1+e_2$ and $e_3$. Thus, in this case, $M_1$ is contained in $\mathbb{R}^2$, and $p=2$.
Now let $M_2\subset \mathbb{R}^3$ the curve parametrized by $(\cos(2t),\cos(t),\sin(t))$, shown below. Clearly, $M_2$ is not in a plane, and therefor $p=3$.
Is there a simple way to discriminate between the two cases, which would generalize to more complicated manifolds (say a 6-dimensional manifold in $\mathbb{R}^{16}$)? One way would be to look at the direction (in $\mathbb{R}^n$) of the tangent space at every point of $M$, to see if that explores all the dimensions of $\mathbb{R}^n$. But is there a formula to compute that?
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I don't really get your question.
As a differentiable manifold, $M_2$ is diffeomorphic to a circle (as it is a compact manifold of dimension 1) and then is embeddable in $\mathbb{R^2}$.