View source for Weingarten surface

A surface whose [[Mean curvature|mean curvature]] is in functional relationship with its [[Gaussian curvature|Gaussian curvature]]. A necessary and sufficient condition for a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097630/w0976301.png" /> to be a Weingarten surface is for both sheets of its focal set to be superimposed on a surface of revolution, and for the edges of regression (cuspidal edges) of the normals to the lines of curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097630/w0976302.png" /> to be superimposed on the meridians. Examples of Weingarten surfaces include surfaces of revolution and surfaces of constant mean or constant Gaussian curvature. Introduced by J. Weingarten [[#References|[1]]], [[#References|[2]]] in the context of the problem of finding all surfaces isometric to a given surface of revolution. The latter problem can be reduced to finding all Weingarten surfaces of this class.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Weingarten,   "Ueber eine Klasse auf einander abwickelbarer Flächen"  ''J. Reine Angew. Math.'' , '''59'''  (1861)  pp. 382–393</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Weingarten,   "Ueber die Flächen, derer Normalen eine gegebene Fläche berühren"  ''J. Reine Angew. Math.'' , '''62'''  (1863)  pp. 61–63</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Shulikovskii,   "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,   "Lectures on classical differential geometry" , Addison-Wesley  (1950)</TD></TR></table>