Surface

A Riemann manifold of dimension 2. That is, a differentiable manifold of dimension 2 together with a Riemannian metric.

Important facts

• Closed surface: in ${\mathbb{R}}^3$, those which correspond to the boundary of a solid object (see @needham2021visual page 165).
• Closed and oriented surface: Mobius realized in 1893 that every closed orientable surface is homeomorphic to a $g$-holed torus, being $g$ the genus.
• Locally they can be isometrically embedded into ${\mathbb{R}}^3$ under the assumption of analyticity. See Cartan-Janet theorem.