# Is a parameterization defined to be surjective and/or injective?

A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know or list here.

1. I was wondering if a parameterization is defined to be surjective? I guess yes, because, in statistics, it seems that if a parameterization is injective, we can then take its inverse. I am not sure if my understanding is correct, and if it is the same case in other areas than statistics.

2. Is it not defined to be injective? I think yes, because for example, in statistics, there is another concept identifiability for an injective parameterization and unidentifiability for an noninjective parameterization. I am not sure if my understanding is correct, and if it is the same case in other areas than statistics.

Thanks and regards!

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In the manifold context, injective only (unless the manifold in question is an open set of $\mathbb R^n$).
When you have an open set of $\mathbb R^n$, it parametrizes itself by the identity map. Otherwise you must cover upba manifold by various pieces, each of which is the bijective image of such an open subset. –  Ted Shifrin Apr 28 '13 at 1:19