AI 中文总结
本文利用移动曲面微积分，给出霍普夫积分曲率定理关键部分的初等证明，证明总曲率与超曲面形状无关，其变化率在超曲面光滑形变下消失。
AI 中文摘要
我们给出霍普夫积分曲率定理关键部分的一个初等证明，即总曲率（高斯-克罗内克曲率B的曲面积分）与超曲面的形状无关。我们利用移动曲面微积分完成此项工作，证明总曲率的变化率在超曲面的光滑形变下消失。
英文摘要
We provide an elementary proof of the key aspect of Hopf's curvatura integra theorem. Namely, we show that the total curvature, i.e. the surface integral of the Gauss-Kronecker curvature B, is independent of the shape of the hypersurface. We accomplish this task by using the Calculus of Moving Surfaces to show that the rate of change of the total curvature vanishes under smooth changes in shape.
