nLab surface

Idea

A surface is a space of dimension 2.

In differential geometry this means a 2-dimensional smooth manifold or something thereby parametrized?.

In complex analytic geometry this usually means a complex manifold of complex dimension 2.

Similarly and more generally, in algebraic geometry an algebraic surface is a variety of algebraic dimension 2.

Examples

• plane

• sphere

• torus

• Klein bottle

• 3-manifold, 4-manifold, 8-manifold

Related concepts

• differential geometry of curves and surfaces

• analog for dimension 1: curve, algebraic curve

• genus of a surface

• area

• moduli space of curves

• complex surface

• surface knot

• hypersurface

• minimal surface

• 3-manifold, 4-manifold, 8-manifold

References

• Manifold Atlas, 2-manifolds

Discussion of de Rham cohomology of surfaces:

• William Fulton, §18 of: Algebraic Topology – A First Course, Graduate Texts in Mathematics 153, Springer (1995) [doi:10.1007/978-1-4612-4180-5]