Generalized polar transforms of spacelike isothermic surfaces

Abstract

In this paper, we generalize the polar transforms of spacelike isothermic surfaces in $Q_1^4$ to n-dimensional pseudo-Riemannian space forms $Q_r^n$. We show that there exist $c-$polar spacelike isothermic surfaces derived from a spacelike isothermic surface in $Q_r^n$, which are into $S_r^{n+1}(c)$, $H_{r-1}^{n+1}(c)$ or $Q_r^n$ depending on $c>0,<0,$ or $=0$. The $c-$polar isothermic surfaces can be characterized as generalized $H-$surfaces with null minimal sections. We also prove that if both the original surface and its $c-$polar surface are closed immersion, then they have the same Willmore functional. As examples, we discuss some product surfaces and compute the $c-$polar transforms of them. In the end, we derive the permutability theorems for $c-$polar transforms and Darboux transform and spectral transform of isothermic surfaces.

12 pages, to appear in J. Geom. Phy

References in corpus (5)

• The submanifold geometries associated to Grassmannian systems
• Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems
• Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms
• Adjoint Transform of Willmore Surfaces in $n$-sphere
• Polar transform of Spacelike isothermic surfaces in 4-dimensional Lorentzian space forms

Cited by in corpus (2)

• On Willmore surfaces in S^n of flat normal bundle
• On transforms of timelike isothermic surfaces in pseudo-Riemannian space forms