Difference between parameterization and embedding of manifolds

Question

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?

Answer 1

A manifold can be given abstractly, as a topological space with additional structure. It can also be given concretely, as a submanifold of $\mathbb R^n$ or some other familiar space (sphere, hyperbolic space...)

Embedding means a homeomorphism (or diffeomorphism, if you care about smoothness) from an abstract manifold onto a submanifold of a familiar space. The emphasis here is on the supposed benefit of realizing an abstract thing within a larger, but concretely given thing.

Parametrization is a homeomorphism (or diffeomorphism, if you care about smoothness) between two manifolds, one of which is a model manifold, such as the standard sphere of unit radius. So, given a smooth strictly convex surface in $\mathbb R^3$, we can use the Gauss map to obtain its parametrization by the unit sphere. One would not usually call this an embedding because we already had a submanifold of $\mathbb R^3$ to begin with. The emphasis here is on the supposed benefit of putting a complicated object in an isomorphism with a simple one.