Let $M,\,N$ be smooth manifolds and $f:\,M\longrightarrow N$ be a smooth map. The rank of $f$ at a point $p\in M$ is defined by the rank of the differential map $df_p$, that is the dimension of $\,df_p(TM_p)\leq TN_{f(p)}$. If the rank of $f$ is constant at every point of $M$, then we say $f$ has constant rank on $M$.
I was interested in the fact that despite the smoothness of the manifolds and the map, the dimension of the image of the tangent space at a point through the differential map needs not to be the same at another point. May you provide me some example for such maps in the case for Euclidean space or in arbitrary manifold ? Thanks.
Trivial examples can be obtained for non-connected $M$. Take e.g. $M = (0,1) \cup (1,2), N = \mathbb R$ and $f(x) = x$ for $x \in (0,1)$, $f(x) = 0$ for $x \in (1,2)$ Then the rank of $f$ is $1$ at $x \in (0,1)$ and $0$ at $x \in (1,2)$.
An example with connected $M$ is $$f : \mathbb R \to \mathbb R, f(x) = x^3 .$$ The rank of $f$ is $1$ at $x \ne 0$ and $0$ at $x = 0$.
You can generalize this to $$f : \mathbb R^n \to \mathbb R^n, f(x_1, \ldots, x_n) = (x_1^3. \ldots, x_n^3) .$$ The Jacobian at $x = (x_1, \ldots, x_n) $ is the diagonal matrix with diagonal entries $$\frac{\partial f_i}{\partial x_i}(x) = 3x_i^2 .$$
The rank of this matrix attains all values $k \in \{0, 1, \ldots, n\}$. The value $k$ occurs precisely at the points $x$ having $k$ non-zero coordinates.