Constant Rank Theorem for Manifolds with Boundary

Solution 1:

Lee's Introduction to Smooth Manifolds deals with the case of local immersions for manifolds with boundary in Theorem 4.15. So let us suppose that $F:\mathbb{H}^m \rightarrow \mathbb{R}^n$ has $\mathrm{rank}(F)=k<m$. For $\mathbb{H}^m=\{(x^1,\dots,x^m)\in \mathbb{R}^m, x^m\geq 0\}$, the assumption $\ker dF_p\not\subseteq T_p\partial\mathbb{H}^m$ in Lee's answer to the original question means that $dF_p(\partial/\partial x^m+a_1\partial/\partial x^1+\dots+a_{m-1}\partial/\partial x^{m-1})=0$ for some numbers $a_i$. The search for $k$ linearly independent tangent vectors in the image can therefore be restricted to $dF_p(\partial/\partial x^i), i<m$. Let's suppose that $k$ is the rank of $(\partial F_i/\partial x^j)_p, 1\leq i, j \leq k$. The coordinate change $(x^1,\dots,x^m) \rightarrow (F_1,\dots,F_k,x^{k+1},\dots,x^m)$ produces another boundary chart $x^m\geq 0$ for which the rest of the proof works as in the case of manifolds without a boundary.