Smale-Hirsch makes rank reduction sufficient for Euclidean immersion in positive codimension

Define $F:=\phi {\mathrm{\mid }}_{TM}:TM\to {\epsilon }^{m+k}$ as the restriction of the bundle isomorphism $\phi$ to the first summand; this is a smooth bundle map over $M$, injective on every fibre. By [F4], it determines a smooth fibrewise injective map $TM\to c^{\ast }T{\mathbb{R}}^{m+k}$ over ${\mathrm{id}}_M$. Composing with the canonical map $c^{\ast }T{\mathbb{R}}^{m+k}\to T{\mathbb{R}}^{m+k}$ gives a smooth bundle map $\tilde{F}$ over $c$, so $(c,\tilde{F})$ is a formal immersion by [F1]. Thus $\mathrm{FImm}(M,{\mathbb{R}}^{m+k})$ is nonempty.

By [F2] with $N={\mathbb{R}}^{m+k}$ (boundaryless, positive codimension $k\ge 1$) the derivative map $D$ is a weak homotopy equivalence; by [F3] it induces a bijection on path components. Since $\mathrm{FImm}(M,{\mathbb{R}}^{m+k})$ is nonempty by step 1.1 and ${\pi }_0(D)$ is surjective, the target's empty-or-non-empty status matches the source's, so $\mathrm{Imm}(M,{\mathbb{R}}^{m+k})$ is nonempty: there exists a smooth immersion $M\looparrowright {\mathbb{R}}^{m+k}$.

Conversely, every smooth immersion $M\looparrowright {\mathbb{R}}^{m+k}$ of the closed $M$ has a rank-$k$ normal bundle which is a rank-$k$ stable normal inverse by [F5], under the same ${\mathrm{A}\mathrm{C}}_{\omega }$. Therefore for closed $M$ and $k\ge 1$ the existence of an immersion into ${\mathbb{R}}^{m+k}$ is equivalent to the existence of a rank-$k$ stable normal inverse: reduction of the structure problem to the normal bundle is sufficient as well as necessary, which is the completeness statement of the design. The argument selects no immersion canonically (it only proves nonemptiness of a space), asserts nothing about the regular homotopy class of the immersion produced, and makes no claim about its normal bundle; the only choice used is the ${\mathrm{A}\mathrm{C}}_{\omega }$ assumed by the Smale-Hirsch theorem and by the normal-bundle splitting.