Let $i:N\to M$ be an injective immersion. $(N,i)$ is called an immersed submanifold of $M$. In this case, $i(N)$ is given a manifold structure from the smooth structure of $N$.
Now consider the inclusion map $i: i(N)\to M$. Is it true that $(i(N),i)$ is an immersed submanifold of $M$?
This question came to my mind when I verified that a nonvanishing integral curve with the inclusion map is an immersed submanifold
Best Answer
To talk about an immersion you need to have a map $j:M_1\to M_2$ where $M_1$ and $M_2$ are manifolds. Here you have a map $i:i(M)\to N$ but $i(M)$ need not be a manifold so it doesn't make sens to be an immersion in this case. So the answer is no.
Edit: I went a bite fast here. Lets call $S=i(M)$ with the differential structure given by $M$. Lets call $j:S\to N$ the map you are looking at. Then $h:M\to S$ defined by $h(m)=i(m)$ is a diffeomorphism (almost by definition). The map you are looking for is just $j=i\circ h^{-1}$, which is an immersion.