Difference between smooth surface and regular surface

Question

I have a question about surfaces. What is the exact difference between regular surface and smooth surface? As in differential geometry they both defined as equivalent existing of tangent plane at every point.

Answer 1

There is no bijection between concepts and terminology. In that sense, unfortunately, the question is mu.

Nonetheless, there are only a few conditions generally intended when someone speaks of smooth surfaces or regular surfaces:

• An abstract smooth $2$-manifold, probably Hausdorff and second-countable. (For brevity let's use the term surface for this.)
• A surface smoothly embedded in a smooth manifold of dimension at least $3$.
• A surface smoothly immersed in a smooth manifold of dimension at least $3$.
• Any of the preceding, equipped with a Riemannian metric. (If the surface has an immersion or embedding, the ambient space is probably assumed to have a Riemannian metric as well, and the surface metric is induced by the ambient metric, see especially Moishe Kohan's answer here.)

The differential-geometry tag suggests the context is a surface smoothly embedded in Euclidean three-space, equipped with the induced metric.

One other note:

• A mapping from an open subset of a Cartesian space to a Cartesian space is smooth when it has derivatives, of all orders or up to some finite order (often depending on the author's standing convention).
• A smooth mapping is regular if its differential is injective at each point, i.e., if the mapping is an immersion.

The types of behavior that cause writers to distinguish smooth from regular in these senses include:

• The mapping $f(u, v) = (\cos^{3} u, \sin^{3} u, v)$ is smooth (infinitely differentiable), and in fact real-analytic, but not regular: The differential has rank $1$ where $u$ is an integer multiple of $\pi/2$. The image, moreover, is not a surface: There are four lines where cross sections $z = \text{constant}$ of the image have cusps.
• The mapping $g(u, v) = (u^{3}, v^{3}, 0)$ is again real-analytic, in fact polynomial, but not regular: The differential has rank $0$ at the origin, and rank $1$ along other points of the coordinate axes. The image, however, is the plane $z = 0$, which is as smooth a surface as can be imagined.