 Smooth surface

In the abstract sense, a smooth surface is a two-dimensional smooth manifold. Its underlying topological space is a topological surface, while its smooth coordinate changes are additional data. No ambient Euclidean space is part of this definition.

In the embedded usage, a regular surface $S\subseteq {\mathbb{R}}^3$ is a two-dimensional embedded submanifold. Locally it has a smooth parametrization $P(u,v)$ with linearly independent tangent vectors $P_u,P_v$, equivalently $P_u\times P_v\ne 0$. The ambient Euclidean metric then induces a Riemannian metric on $S$.

An abstract surface can be supplied with a smooth embedding into some ambient Euclidean space, but the embedding and its induced metric are extra choices. A parametric surface can also be only an immersion: local rank two does not ensure global injectivity or the subspace topology. The irrational winding of the torus illustrates the analogous distinction for curves, and the dense immersed cylinder in a three-dimensional torus illustrates it for surfaces.