Surfaces

A surface is a two-dimensional manifold. In differential geometry, a smooth surface is a two-dimensional differentiable manifold that can be embedded in three-dimensional space; locally each point has a neighborhood diffeomorphic to the plane. Surfaces may be abstract, defined by their topology, or concrete as 2D subsets of a higher-dimensional space with a smooth structure.

Classic examples include the plane, the sphere, and the torus. Some surfaces are non-orientable, such as the Klein bottle and the real projective plane. Compact surfaces are classified by orientability and genus: orientable surfaces are homeomorphic to a sphere with g handles; non-orientable surfaces are connected sums of k projective planes.

The geometry of a surface is encoded by curvature. For a smooth surface in R^3, Gaussian curvature K varies across the surface. The Gauss-Bonnet theorem links the total curvature to the topology through the Euler characteristic.

Special types include minimal surfaces (mean curvature zero), ruled surfaces (swept by lines), and developable surfaces (zero Gaussian curvature along infinitesimal neighborhoods; examples are planes, cylinders, cones). Surfaces of revolution are formed by rotating a plane curve. In applied fields, surfaces arise in computer graphics, physical modeling, and architecture.