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WEINGARTEN SURFACE

Notion defined by Weingarten in 1861 then studied
by Hopf in 1951.
Julius Weingarten (1836-1910): German mathematician. Link: www.encyclopediaofmath.org/index.php/Weingarten_surface REF : [KÜHNEL] page 93 |

A surface is said to be a Weingarten surface if there
exists a relation, that does not depend on the parameters, between the
mean
curvature and the total curvature* *(or between the principal
curvatures).

Examples : obviously, the surfaces with constant mean
or total curvature (of witch the developable
surfaces), but also all toutes les surfaces
of revolution : éliminate between
both expressions of the curvatures ; for example, the ellipsoid of revolution
presents the relation
between it's curvatures.

Theorem of Beltrami and Dini (1865) : the ruled surfaces
that are of Weingarten are the developable
surfaces and the ruled
helicoids.

Theorem
of Voss (1959) : the compact analytic surfaces of genus
0 that are of Weingarten are rotation surfaces; see for example the case
where the
curvatures are proportional.

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© Robert FERRÉOL 2019