On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space

Folding maps on a cross-cap

We study the reflectional symmetry of a surface in the Euclidean 3-dimensional space with a cross-cap singularity with respect to planes. This symmetry is picked up by the singularities of folding maps on the cross-cap. We give a list of the generic singularities that appear in the members of the family of folding maps on a cross-cap and characterise them geometrically.

Curved foldings with common creases and crease patterns

Consider a curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $Γ$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilit…

实解析映射芽的 $\rho$-管和 $\rho$-球上的 Milnor 纤维化

Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs

本文研究实解析映射芽在替换欧氏距离为解析控制函数后的Milnor纤维化，给出管和球纤维化的判据，并通过例子表明控制函数的选择可决定纤维化是否存在。

Circles and Clifford Algebras

Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between our result and the Hurwitz--Radon theorem about sums of squares.

Circles and Quadratic Maps Between Spheres

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has rank at least 2, is equivalent to a fractional quadratic rounding. A fractional quadratic map is just the ratio of a quadratic map and a quadratic polynomial. We also show that any rounding gives rise to a quadratic map between spheres. The known results on quadratic maps between …

Lorentz-Minkowski 3-空间中曲面的脐点

Umbilics of surfaces in the Lorentz-Minkowski 3-space

本文研究了Lorentz-Minkowski 3-空间中类空与类时曲面脐点的基本性质，证明了类空曲面脐点局部行为与欧氏空间一致，并给出了类时曲面孤立脐点指数的零与非零条件。