"Parametric" versus "Non-parametric" hypersurface

Question

What exactly are the meanings of the terms "parametric hypersurface" and "non-parametric hypersurface"?

My initial guess was "parametric" referred to a hypersurface that is a global graph (e.g. the graph of a parabola in $\mathbb{R}^{2}$ or the graph of the tangent function in $(-\pi/2,\pi/2) \times \mathbb{R}$) --- graphs are explicitly "parametrized" ---, whereas "non-parametric" referred to hypersurfaces that might not be graphs everywhere, like the circle --- which do not have a "canonical parametrization." However, I fear it's the exact opposite --- and I don't understand the reasoning behind the terminology.

The terminology seems to be taken for granted in differential geometry papers (e.g. this and this).

Answer 1

"Parametric" or "parameterized" means given as an image of a map from a domain $\mathbb{R}^2$ (usually domain="a non-empty open set"). Thus a parametric surface is one that is given as an image of a map, say $x(u,v) = u(1 - u^2/3 + v^2)/3, y(u,v)= v(1 - v^2/3 + u^2)/3, z(u,v)= (u^2 - v^2)/3.$ This does not have to be graphical.

Now, a graph of a function $f(x,y)$ has a standard parametrization $x=u, y=v, z=f(u,v)$. However, one does not need to explicitly invoke this parametrization, since the surface is now encoded in the function $f$. Thus, in the minimal surfaces literature, the surface is said to be given in "non-parametric form". (In higher simensions, we have a graph of $n-2$ functions $x(x_1,x_2) = (x_1,x_2,f_3(x_1,x_2),... ,f_n(x_1,x_2))$.) Of course, any regular surface can be locally given in such a form.

All of this is explained, in particular, in the first chapter of Oseerman's "A Survey of Minimal Surfaces".

This terminology is indeed potentially confusing, but I believe it stems from the fact that the methods of study of such non-parametric surfaces focus more on the function $f$, and thus are different from the methods employed in the parametric case.

Answer 2

Most differential topologists will avoid this terminology, but it is used by researchers in Differential Geometry. In the context of papers you are reading, this distinction amounts to the following:

1. A non-parametric hypersurface is simply a smooth codimension 1 submanifold.

2. A parametric hypersurface in an $m$-dimensional smooth manifold $M$ is a pair $(N, f)$, where $N$ is a smooth $m-1$-dimensional manifold and $f: N\to M$ is a smooth embedding. (A differential topologist in this situation will simply say "a codimension 1 embedding" instead of a "parametric hypersurface".) Sometimes people get sloppy and use this terminology even if $f$ is not an embedding, so read closely.

In the context of geometric flows studied by differential geometers, in the non-parametric case they will consider a family of codimensional one submanifolds $N_t\subset M$ (say, given by the mean curvature flow), while in the parametric case they would consider either a family of maps $f_t: N\to M$ or a family of maps $f_t: (N,g_t)\to M$, where $g_t$ is a family of Riemannian metrics on $N$. For instance, such flows appear when constructing a minimal map $f_0: (N,g_0)\to M$, where $M$ is given a Riemannian metric and $N$ is a surface. This approach was used, for instance, by Rado to solve the classical Plateau problem.