I´ve a question involving two definitions:
For $M$ a differential manifold, is $R \subset M$ a submanifold if the inclusion $i : R \longrightarrow M$ is an immersion (that is, is differentiable with $\operatorname{rank}(i) = \dim M$).
Also, $R \subset M$ a regular submanifold if the inclusion $i : R \longrightarrow M$ is an immersion and also $i : R \longrightarrow i(R) \subset M$ is an homeomorphism.
Thanks.
Note: But any inclusion $i : R \to i(R) \subset M$ is an homeomorphism, yeah? So, any submanifold is a regular manifold.
The second definition is stronger than the first one. Sometimes, such a manifold is called embedded submanifold. It means that the submanifold topology on it is the same as the subspace topology.