论文标题
Corank的轴向曲率1单数$ n $ -manifolds in $ \ mathbb r^{n+k} $
Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb R^{n+k}$
论文作者
论文摘要
对于单数$ n $ -manifolds，in $ \ mathbb r^{n+k} $，带有corank 1的单个点，$ p \ in m^n _ {\ mbox {sing}} $我们最多定义$ l（n-1）$ $ l（n-1）$不同的轴向curvatures $ p $ p $ p $ p $，其中$ l = \ l = \ f = \ f = \ min min \ min \米\ \ k+{这些曲率是使用曲率基因座（通过单位切线向量的第二个基本形式的图像）获得的，因此是二阶不变的。实际上，在$ n = 2 $的情况下，它们概括了针对额叶类型表面定义的所有二阶曲率。我们将这些曲率与主要曲率在$ m^n _ {\ mbox {sing}} $中包含的相关常规$（n-1）$的某些正常方向上相关联。在$ n = 2,3 $的情况下，我们获得了许多有趣的几何解释。例如，对于具有二维单数集的额叶型3型脉冲，可以用轴向曲率表示单数集的高斯曲率。同样，对于奇异集的曲率是1维集合时的曲率。最后，我们表明，到目前为止为奇异歧管定义的所有脐带曲率都可以看作是我们轴向曲率之一的绝对值。
For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singular point at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axial curvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained using the curvature locus (the image by the second fundamental form of the unitary tangent vectors) and are therefore second order invariants. In fact, in the case $n=2$ they generalise all second order curvatures which have been defined for frontal type surfaces. We relate these curvatures with the principal curvatures in certain normal directions of an associated regular $(n-1)$-manifold contained in $M^n_{\mbox{sing}}$. We obtain many interesting geometrical interpretations in the cases $n=2,3$. For instance, for frontal type 3-manifolds with 2-dimensional singular set, the Gaussian curvature of the singular set can be expressed in terms of the axial curvatures. Similarly for the curvature of the singular set when it is 1-dimensional. Finally, we show that all the umbilic curvatures which have been defined for singular manifolds up to now can be seen as the absolute value of one of our axial curvatures.
