View source for Isothermal surface

A surface whose curvature lines form an [[Isothermal net|isothermal net]]. For example, quadrics, surfaces of rotation, surfaces of constant mean curvature, and, in particular, minimal surfaces are isothermal surfaces (cf. [[Quadric|Quadric]]; [[Rotation surface|Rotation surface]]; [[Minimal 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|Peterson correspondence]] with it. [[Inversion|Inversion]] of space preserves the class of isothermal surfaces.



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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''5''' , Publish or Perish  (1975)  pp. 1–5</TD></TR></table>