Immersion

Remarks

Equivalently, the rank of $df_p$ equals $\dim M$ at every point $p$; in particular $\dim M \le \dim N$. An immersion that is a homeomorphism onto its image, with the subspace topology, is a smooth embedding. Example: $f \colon \R \to \R^2$, $f(t) = (\cos t, \sin t)$, is an immersion. Its derivative $f'(t) = (-\sin t, \cos t)$ never vanishes, so $df_t$ is injective for every $t$. It is not an embedding, because it is not injective. Non-example: $g(t) = (t^2, t^3)$ is smooth and injective but is not an immersion. Its derivative $g'(t) = (2t, 3t^2)$ vanishes at $t = 0$, where the image has a cusp. By the constant rank theorem, near each point an immersion has the local form $(x_1, \dots, x_m) \mapsto (x_1, \dots, x_m, 0, \dots, 0)$ in suitable charts.