主曲率曲面上的几何
Geometry on principal curvature surfaces
浏览论文内容
中文总结 AI 辅助
研究与惠特尼伞相关的主曲率曲面几何，证明其与法平面交线和焦圆锥性质，通过研究多种曲率沿例外集的情况，确定曲率函数零点数量。
中文摘要 AI 辅助
我们研究与惠特尼伞相关的主曲率曲面的几何。该曲面通过沿法向以其有界主曲率扩展惠特尼伞得到，具有自然几何解释。我们证明它与法平面的交线与曲率抛物线的垂足曲线重合，并推断焦圆锥是该垂足曲线的反演。然后通过研究其测地线和法曲率以及高斯曲率和平均曲率，研究主曲率曲面沿例外集的几何，并确定这些曲率函数零点的所有可能一般数量。
英文摘要
We study the geometry of the principal curvature surface associated with a Whitney umbrella. This surface is obtained by extending the Whitney umbrella in the normal direction by its bounded principal curvature and admits a natural geometric interpretation. We prove that its intersection with the normal plane coincides with the pedal curve of the curvature parabola and deduce that the focal conic is the inversion of this pedal curve. We then investigate the geometry of the principal curvature surface along the exceptional set by studying its geodesic and normal curvatures, as well as its Gaussian and mean curvatures, and determining all possible generic numbers of zeros of these curvature functions.
