论文标题
高维欧几里德空间中不规则曲线的弱曲率
Weak curvatures of irregular curves in high dimension Euclidean spaces
论文作者
论文摘要
我们处理在高维空间中定义的一系列不规则曲线的较弱正态概念。关于多边形曲线，离散的正态是通过应用于连续定向段的革兰氏schmidt程序来构建的，它们自然地生活在与高斯超级球员相关的投射空间中。通过使用具有无限模量的铭文多边形的序列，然后引入了$ j $的总变化的放松概念 - 与通用曲线的往来。对于满足约旦系统的平滑曲线，实际上，我们的放松概念与平滑的$ j $ th的长度一致。 相应地，引入了有限松弛能量的不规则曲线的较弱$ j $的良好概念，事实证明，这是任何近似多边形的任何序列的强限制。 我们弱正常的长度与相应的松弛能量一致，还可以获得相关的积分几何公式。 然后，我们讨论一类更广泛的平滑曲线，该曲线弱的曲线与拐点之外的经典曲线严格相关。 最后，从弱$ j $的长度的第一个变化开始，也分析了曲率度量的自然概念。
We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram-Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated to the Gauss hyper-sphere. By using sequences of inscribed polygonals with infinitesimal modulus, a relaxed notion of total variation of the $j$-th normal to a generic curve is then introduced. For smooth curves satisfying the Jordan system, in fact, our relaxed notion agrees with the length of the smooth $j$-th normal. Correspondingly, a good notion of weak $j$-th normal of irregular curves with finite relaxed energy is introduced, and it turns out to be the strong limit of any sequence of approximating polygonals. The length of our weak normal agrees with the corresponding relaxed energy, for which a related integral-geometric formula is also obtained. We then discuss a wider class of smooth curves for which the weak normal is strictly related to the classical one, outside the inflection points. Finally, starting from the first variation of the length of the weak $j$-th normal, a natural notion of curvature measure is also analyzed.
