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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^{2 }
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.

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© Robert FERRÉOL 2019