Is the smooth map with constant rank dense?

Question

Let $M$ and $N$ be two Riemannian manifold. Assume that $f:M\rightarrow N$ is a smooth map and $\dim M < \dim N$. Can we find a smooth map $g$ $C^{k}$-close to $f$ which is immersive at each point $p\in M$, i.e., the induced map on the tangent space $T_pM$ is an injection?

Answer 1

In some cases, yes. The following is of Theorem 2.2.12 in Hirsch's "Differential Topology" book.

Let $M,N$ be $C^k$ manifolds, $1 \leq k \leq \infty$. If $\dim N \geq 2\dim M$, then immersions are dense in $C^k_S(M,N)$, ths space of $C^k$ maps $M \to N$ equipped with the strong topology.

You needn't be able to do much better than this, as it's possible that $M$ does not immerse into $N$ at all. For instance, by a Stiefel-Whitney class argument $\Bbb{RP}^{2^n}$ does not immerse in $\Bbb R^m$ for $m < 2^{n+1}-1$. (In particular, $\Bbb{RP}^4$ first immerses in $\Bbb R^m$ for $m=7$.)

I would suspect that you can't do better at all (ie, there's $M,N$ such that $\dim N = 2\dim M - 1$ and immersions are not dense), since one has to work hard to actually immerse every $n$-manifold into $\Bbb R^{2n-1}$; it's not a 'general position' argument, unlike the argument showing we can immerse every $n$-manifold into $\Bbb R^{2n}$.