Regular Surface

A Subset $M\subset\Bbb{R}^n$ is called a regular surface if for each point $p\in M$, there exists a Neighborhood $V$ of $p$ in $\Bbb{R}^n$ and a Map $x:U\to \Bbb{R}^n$ of an Open Set $U\subset \Bbb{R}^2$ onto $V\cap M$ such that

1. $x$ is differentiable,

2. $x:U\to V\cap M$ is a Homeomorphism, and

3. Each map $x:U\to M$ is a Regular Patch.

Any open subset of a regular surface is also a regular surface.

See also Regular Patch

References

Gray, A. ``The Definition of a Regular Surface in $\Bbb{R}^n$.'' §10.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 195-200, 1993.