Isothermal surface

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A surface whose curvature lines form an isothermal net. For example, quadrics, surfaces of rotation, surfaces of constant mean curvature, and, in particular, minimal surfaces are isothermal surfaces (cf. Quadric; Rotation surface; Minimal surface). An invariant criterion for a surface to be isothermal is that the Chebyshev vector of the net of curvature lines is gradient. For each isothermal surface one defines another isothermal surface which is, up to a homothety, in conformal Peterson correspondence with it. Inversion of space preserves the class of isothermal surfaces.



[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 2 , Gauthier-Villars (1887) pp. Chapt. XI, no. 429ff
[a2] M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a3] M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5
How to Cite This Entry:
Isothermal surface. I.Kh. Sabitov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098