SURFACE

It is a very good question, and I thank you even less (see curve) for asking it

Like for the curves, we cannot define a surface in a general fashion, but only various aspects of this notion. We will only deal here with the topological aspect.

A topological surface is a topological space locally homeomorphic to the plane or the half-plane (i.e. all points of which have a neighborhood homeomorphic to R or ); it is a topological manifold of dimension 2.

The points that have a neighborhood homeomorphic to the half-plane constitute the "boundary" of the surface; a surface without boundary is said to be simple.

It can be proved that any simple compact connected (or closed) surface is homeomorphic either to a connected sum of tori if it is orientable, or to the connected sum of real projective planes otherwise.

The surfaces can be classified according to their geometry:
- Euclidian, if by a point there passes a unique line (=geodesic) parallel to a given line, such as the plane (we also talk about a "flat" surface)
- spherical, if by a point there does not pass any line parallel to a given line, such as the sphere
- hyperbolic, if by a point there pass an infinite number of lines parallel to a given line, such as the hyperboloid of revolution.

The Euclidian closed surfaces are those with mean Gaussian curvature equal to zero and therefore, according to the Gauss-Bonnet formula, those with Euler characteristic equal to zero: the torus and the Klein bottle.

The spherical closed surface are those with positive mean Gaussian curvature, therefore, those with positive Euler characteristic: the sphere and the projective plane.
All the others are hyperbolic.