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TIGHT SURFACE
Surfaces studied by 
An orientable closed surface is said to be tight if its total absolute Gauss curvature is minimal among the surfaces of the same genus (this total curvature being defined by where is the Gauss curvature, product of the principal curvatures).
This minimal value is given by the formula where g is the genus of the surface and its EulerPoincaré characteristic.
The inequality with equality if the surface is tense, is to be compared with the equality of GaussBonnet: .
The following beautiful theorem concretely characterizes tight surfaces [Kühnel, p. 186] :
For a closed orientable surface S, noting the portion of S having a positive Gauss curvature, the following conditions are equivalent:
a) S is tight.
b)
c)
d) Any plane separates S into two or less connected
components.
In particular the 0genus tight surfaces are the boundary
surfaces of the convex bodies of .
Geometric tori are tight surfaces, because any plane separates them into two or less parts, or because.  This 2genus surface is not tight.  This 2genus surface proposed by Banchoff and Kuiper
in 1981 is tight (see [Kühnel,
p. 186]).
It is a component of the surface of equation . 



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© Robert FERRÉOL 2019