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.
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.
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!