Polar Transform of Spacelike Isothermic Surfaces in 4-Dimensional Lorentzian Space Forms

2016

Results Math

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The basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms is introduced, which is parallel to the classical framework of Burstall etc. for spacelike surfaces. Then we provide a discussion on the transforms of timelike (±)−isothermic surfaces (or real isothermic, complex isothermic surfaces), including c− polar transforms, Darboux transforms and spectral transforms. The first main result is that c−polar transforms preserve timelike (±)−isothermic surfaces, which are generalizations of the classical Christoffel transforms. The next main result is that a Darboux pair of timelike isothermic surfaces can also be characterized as a Lorentzian O(n − r + 1, r + 1)/O(n − r, r) × O(1, 1)−type curved flat. Finally two permutability theorems of c−polar transforms are established.

Section: Introductionmentioning

confidence: 94%

“…We define the Darboux transforms and give the basic properties of Darboux transforms as below (compare [7], [18], [28])…”

Section: Darboux Transforms Of Timelike Isothermic Surfacesmentioning

confidence: 99%

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On transforms of timelike isothermic surfaces in pseudo-Riemannian space forms

Song

2016

Results Math

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“…(We also know that any of such tori is isothermic, which means that the conformal Hopf differential is real valued with respect to a suitable chosen coordinate z. For a discussion about such surfaces in Q 4 1 , see [20][21][22].) In particular, the left adjoint surface of Y is given by…”

Section: R] ([L]) Is a Point Using The Conclusion In (I) For Surfacementioning

confidence: 99%

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Sci. China Ser. A-Math.

2008

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Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

“…On the other hand, in [12,13], Ma and Wang found that there exists a natural transform of spacelike surfaces in Q 4 1 , the conformal compactification of the 4-dimensional Lorentzian space forms. The key observation is that in this codim-2 case, the normal plane at any point is Lorentzian.…”

Section: Introductionmentioning

confidence: 99%

Generalized polar transforms of spacelike isothermic surfaces

Journal of Geometry and Physics

2012

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In this paper, we generalize the polar transforms of spacelike isothermic surfaces in $Q^4_1$ to n-dimensional pseudo-Riemannian space forms $Q^n_r$. We show that there exist $c-$polar spacelike isothermic surfaces derived from a spacelike isothermic surface in $Q^n_r$, which are into $S^{n+1}_r(c)$, $H^{n+1}_{r-1}(c)$ or $Q^n_r$ 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.Comment: 12 pages, to appear in J. Geom. Ph