Is it always true that a local diffeomorphism is an immersion? I was thinking about the specific case of the equivalence map:
$$ \rho : (x,y,z) \longrightarrow [(x,y,z)]$$
that goes from $S^2$ to $P(\mathbb{R^2})$, the latter being the projective space for $n=2$. We know that this is a local diffeomorphism and that the sphere is compact but does that allow us to conclude that it is also an immersion? Or is it not an immersion?
This is trivial with certain definitions of "immersion" and "local diffeomorphism". Let $f$ be a smooth map between manifolds.
What it means for $f$ to be an immersion is that $df_x$ is injective for all $x$.
What it means for $f$ to be a local diffeomorphism is that $df_x$ is bijective for all $x$.
Thus, a local diffeomorphism is always an immersion.
Another definition of "local diffeomorphism" that you may have seen is "for all points $x$, there is an open neighborhood $U$ of $x$ such that $f|_U$ is a diffeomorphism onto its image". The inverse function theorem says precisely that this is equivalent to the definition of "local diffeomorphism" given above.