论文标题
4维空间形式的引擎盖和等距等温表面
Bonnet and Isotropically Isothermic Surfaces in 4-Dimensional Space Forms
论文作者
论文摘要
我们研究了四维空间形式的表面的引擎盖问题，如果存在平行矢量束等轴测图，则两个等距表面具有相同的平均曲率，以保留平均曲率矢量场。我们处理具有相同平均曲率的等距表面一致性类别的模量空间，并通过其结构在表面上遗传的属性。对这个问题的研究导致我们具有一种新型的共同不变特性，称为同位素等温度，这与完全脐带超曲面的表面的等温度概念相吻合，并且与弯曲和无限等等距变形线有关，以保留平均弯曲矢量。我们表明，如果简单连接的表面不是适当的引擎盖，那么它最多可以接收一个或恰好是三个引擎盖伴侣。对于简单连接的适当的引擎盖表面，模量空间是1维空间，最多可以与圆的连接组件差异，或者是二维圆环。我们证明，完全连接的引擎盖表面位于完全测量的超曲面中，作为非恒定平均曲率的表面，允许不在任何完全脐带超出表面的引擎盖伴侣。我们表明，各向同性等温度表征了适当的引擎盖表面，并且我们为紧凑型表面不存在引擎盖配偶提供了相关条件。此外，我们研究了局部适当的引擎盖的紧凑表面，我们证明在局部模量空间上存在均匀的子结构，其表征既不是最小，也不是超级概念的，呈垂直谐波的高斯升力表面。特别是，我们表明，具有模量空间的唯一紧凑，局部适当的引擎盖表面是圆环的曲线，是那些具有平行平均曲率矢量场和阳性属的圆环。
We study the Bonnet problem for surfaces in 4-dimensional space forms, where two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. We deal with the moduli space of congruence classes of isometric surfaces with the same mean curvature, and with properties inherited on a surface by its structure. The study of this problem led us to a new conformally invariant property, called isotropic isothermicity, that coincides with the usual concept of isothermicity for surfaces lying in totally umbilical hypersurfaces, and is related to lines of curvature and infinitesimal isometric deformations that preserve the mean curvature vector field. We show that if a simply-connected surface is not proper Bonnet, then it admits either at most one, or exactly three Bonnet mates. For simply-connected proper Bonnet surfaces, the moduli space is either 1-dimensional with at most two connected components diffeomorphic to the circle, or the 2-dimensional torus. We prove that simply-connected Bonnet surfaces lying in totally geodesic hypersurfaces as surfaces of nonconstant mean curvature, admit Bonnet mates that do not lie in any totally umbilical hypersurface. We show that isotropic isothermicity characterizes the proper Bonnet surfaces, and we provide relevant conditions for non-existence of Bonnet mates for compact surfaces. Moreover, we study compact surfaces that are locally proper Bonnet, and we prove that the existence of a uniform substructure on the local moduli spaces, characterizes surfaces with a vertically harmonic Gauss lift that are neither minimal, nor superconformal. In particular, we show that the only compact, locally proper Bonnet surfaces with moduli space the torus, are those with nonvanishing parallel mean curvature vector field and positive genus.
