# Weingarten surface

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.

#### References

[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) |

#### Comments

#### References

[a1] | D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950) |

**How to Cite This Entry:**

Weingarten surface.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Weingarten_surface&oldid=32590