Behavior of Gaussian curvature near non-degenerate singular points on wave fronts
Behavior of Gaussian curvature near non-degenerate singular points on wave fronts
复制标题
波前非简并奇点附近高斯曲率的行为
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Kotaro Yamada
中科院分区：
文献类型：
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作者：
L. F. Martins;K. Saji;M. Umehara;Kotaro Yamada
我们定义了黎曼$3$流形中沿尖边的尖曲率$\kappa_c$，并证明它给出了平均曲率函数的发散项的系数。此外，我们证明$\kappa_c$和极限法曲率$\kappa_\nu$的乘积$\kappa_\Pi$（称为乘积曲率）的绝对值是曲面的内在不变量，并且与高斯曲率的有界性密切相关。我们还考虑当尖边累积到其他奇点时 $\kappa_\Pi$ 的限制行为。此外，还给出了尖边和燕尾的几个新的几何不变量。
We define cuspidal curvature $\kappa_c$ along cuspidal edges in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the absolute value of the product $\kappa_\Pi$ (called the product curvature) of $\kappa_c$ and the limiting normal curvature $\kappa_\nu$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $\kappa_\Pi$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.