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