When are two submanifolds "the same"? Consider two smooth submanifolds $N\subseteq\mathbb R^n$ and $M\subseteq\mathbb R^m$. Let there be a function $\varphi\colon N \to M$ that is bijective.

Which properties does the function $\varphi$ need to have so that $N$ and $M$ can be considered the "same manifold". (I guess that means they have the same geometric properties)

If $N$ and $M$ were groups then $\varphi$ would need to be an isomorphism so that $N$ and $M$ are "the same" group. I want something similar for submanifolds.
Since you talk about smooth equivalence, $\varphi$ needs to be a diffeomorphism. A weaker condition is topological equivalence, called homeomorphism.

Judging from your question, you didn't start reading any book on the subject yet. I think it is better to learn a little first and then start to ask questions.
Since you talk about smooth equivalence, $\varphi$ needs to be a diffeomorphism. A weaker condition is topological equivalence, called homeomorphism.

Judging from your question, you didn't start reading any book on the subject yet. I think it is better to learn a little first and then start to ask questions.