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Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold?

  1. $F_0 = f$;
  2. $F_t : M \to \overline{M}$ is an immersion for every $t \in [0,1]$;
  3. $F_1(M)$ has only transverse self-intersections.

(Here, $F_t(p) = F(p,t)$ for every $(p,t) \in M \times [0,1]$.)

If this does not hold in this full generality, is it true for hypersurfaces ($k=1$)? For $\overline{M} = \mathbb{R}^{n+k}$?

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    $\begingroup$ I haven't thought about the proof, but I refer you to Exercise 2 on p. 82 of Hirsch's Differential Topology. The set of proper immersions that are in general position is dense and open in the space of $C^r$ immersions. (His definition of general position is that whenever $x_1,\dots,x_k$ are distinct points with the same image $y$, the tangent space of the ambient manifold at $y$ is spanned by the image of $df_{x_k}$ and the intersection of the images of $df_{x_1},\dots,df_{x_{k-1}}$. $\endgroup$
    – Ted Shifrin
    Commented Aug 29, 2018 at 16:53
  • $\begingroup$ How does one define transverse self intersections ? $\endgroup$
    – Amr
    Commented Oct 17, 2021 at 11:12

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