A subset $N$ of a manifold $M$ which is itself a manifold is a submanifold.
It can be an embedded manifold, if the topology of $N$ is the subset topology of $M$. If not, it is an injective immersed manifold. Keep an eye: if we don't require to be injective, an immersed manifold is not a submanifold!!
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Author of the notes: Antonio J. Pan-Collantes
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