TITLE: What exactly is an embedding in differential geometry?
QUESTION [7 upvotes]: I have seen several definitions of embedding in differential geometry. Let $f:M \to N$ be an injective smooth map between manifolds. Then I have heard that $f$ is an embedding if:

$f_*$ is injective, that is, $f$ is an immersion
for any $p \in M$ there exists a neighbourhood $U$ of $p$ with $f^{-1}:f(U) \to M$ smooth
$f$ is diffeomorphic onto its image
$f$ is an immersion which is also homeomorphic onto its image

My question is: are all these definitions equivalent, are some of them slightly incorrect, or are different definitions relevant for different purposes? Thanks.

REPLY [6 votes]: The definition that I am familiar with is the last definition.  For an embedding, the two properties give different ideas:

A bijective immersion means that there is some topology that can be placed on the image so that the image is homeomorphic to the original manifold.
The condition of being homeomorphic with its image using the subspace topology says that the topology in the bullet above is actually the subspace topology (not some other weird one that differs from $N$'s topology).

Now, some of what you write may be equivalent (certainly not the first one), and the diffeomorphism in the third one needs to be from the subspace topology.  In general, an embedding means that the image is a regular submanifold of the codomain, so $f(M)$ looks like $M$ and also looks like a submanifold of $N$.