Embedding of smooth manifolds

[center o bass]

Consider a smooth map $F: M \to N$ between two smooth manifolds $M$ and $N$. If the pushforward $F_*: T_pM \to T_{F(p)} N$ is injective and $F$ is a homeomorphism onto $F(M)$ we say that $F$ is a smooth embedding.

In analogy with a topological embedding being defined as a map that is homemorphic onto it's image I would think that an embedding of smooth manifolds would require that $F$ be a diffeomorphism. So my question is; is the definition above equivalent with $F$ being a diffeomorphism between $M$ and $N$.

 

[jvicens]

Thank you for your question. To clarify, a diffeomorphism is a smooth map that is both a homeomorphism and a smooth embedding. Therefore, the definition given in the post is indeed equivalent to $F$ being a diffeomorphism between $M$ and $N$. The key difference between a diffeomorphism and a smooth embedding is that a diffeomorphism is invertible, while a smooth embedding may not necessarily be invertible.

In other words, a diffeomorphism is a bijective smooth map with a smooth inverse, while a smooth embedding is only required to be a bijective smooth map with an injective pushforward. This means that a smooth embedding is a weaker condition than a diffeomorphism, and not all smooth embeddings are diffeomorphisms.

I hope this clarifies the relationship between a smooth embedding and a diffeomorphism. Please let me know if you have any further questions.