Definition of a surface

2026-01-12

The definition of surface is taken from Montiel and Ros.

A surface (in $\R^3$) is defined as a nonempty $S \subset \R^3$ such that for each $p \in S$, there is an open $U\subset \R^2$ and an open neighborhood $V \subset S$ (open in the induced topology on $S$), and a differentiable map $X : U \to \R^3$ such that

• $X(U) = V$
• $X: U \to V$ is a homeomorphism
• $(dX)_q : \R^2 \to \R^3$ is injective for all $q \in U$.

The last bullet point makes (a posteriori?) $X$ an immersion. It is useful at the very least to ensure that the tangent planes are always two dimensional: if you define the tangent plane at $X(q) \in V$ to be the span of $X_u(q) \coloneqq \frac{\partial X}{\partial u}$ and $X_{v}(q)$, then the third bullet ensures that these always form a tangent plane.

In general, if $T:V\to W$ is a linear map of finite-dimensional vector spaces, then $T$ injective $\iff \ker T = \{0\} \iff $ dimension of column space is $\dim V$. The second “$\iff$” is a consequence of rank+nullity theorem.

Back to the definition of a surface, the column space of $(dX)_{q}$ is the span of $X_u(q)$ and $X_v(q)$, so by the general remark requiring the third bullet point ensures that tangent spaces will be the right dimension.

All three bullet points make $X$ an embedding, i.e. an immersion which is a homeomorphism onto its image.