Let $i: [0,1]\hookrightarrow\mathbb{R}^3$ be a smooth embedding. Can I find arbitrary small perturbations $j$ of $i$ s.t. $j(0)=i(0)$, $j(1)=i(1)$ and $j'(t)$ is never a multiple of $(0,0,1)$?
More generally, if $i: N\hookrightarrow M\times [0,1]$ is an embedding of (compact if necessary) manifolds, $\dim N\leq \frac{1}{2}\dim M$, can I always find arbitrary small perturbations (wrt. some metric on $M$) $j$ of $i$ s.t.
- $i(x)=j(x)$ whenever $i(x)\in M\times \{0,1\}$
- $j_*(TN)$ never contains a vector $\partial_t$ in the "vertical" direction?
If necessary, consider just the second condition. I will appreciate a hint or reference to this kind of problems. (I need to project an ambient framed cobordism in $M\times [0,1]$ to $M$..)
Update: in the discussion below was proposed to consider the composition $$ j:N\stackrel{0}{\hookrightarrow} TN\stackrel{\iota}{\to} T(M\times [0,1])\stackrel{\pi}{\to} M\times [0,1] $$ with $\iota$ some generic map whose image avoids the vertical distribution. However, the equation $dj=\iota$ doesn't hold in general (or at least I don't see it).