Weingarten surface

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A surface whose mean curvature is in functional relationship with its Gaussian curvature. A necessary and sufficient condition for a surface $S$ 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 $S$ 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 [1], [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.


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



[a1] D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950)
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Weingarten surface. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article