Comprehensive studies of cut locus

我们研究了割轨及其相关的一些问题，如凸曲面上的最远点或简单闭测地线，得到了许多结果，作为确定割轨的问题，我们证明了某些紧致Liouville流形具有一般点的割轨光滑嵌入余维为1的闭圆盘的性质。具有不同轴的椭球是这种流形的典型例子。我们讨论了Jacobi定理（椭球上的共轭轨迹有正四个尖点）的推广，取得了一些显著的进展，并计划继续研究。此外，作为相关课题，我们利用第一积分（与K。作为研究割轨迹结构的问题，在一定的非退化假设下，证明了割轨迹具有良好的分层性，局部具有锥结构。在较强的假设条件下，利用距离函数的临界点，我们得到了莫尔斯理论的简化过程（与T.本文对一般凸曲面建立了简单闭拟测地线的直径、面积和长度不等式（与C。利用上述简单闭拟测地线，我们证明了任何多面体通过某种切割都可以展开为平面简单多边形（与J.O'Rourke，C. Vilcu），并利用简单的Clodes拟测地线讨论了其它几种开折，

We studied the cut loci and related several topics, for example, the farthest points or simple closed geodesies on convex surfaces and got many results.As the problem to determine the cut locus, we proved that some compact Liouville manifolds have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds. We discussed on an extension to general dimension of Jacobi's last theorem(conjugate loci on ellipsoids have exact four cusps), and we got some remarkable progress and are planning to continue the research. Moreover, as the related topics, we got a modern proof of thread constructions of general quadric(hyper)surfaces by using the first integral (jointed work with K. Kiyohara).As the problem to study structures of cut locus, under some non-degenerating assumption we proved that the cut locus admits a nice stratification, some cone structure locally. Under stronger assumptions we have simpler procedure of Morse theory by using of critical points of distance functions (jointed work with T. Sakai).We established, for general convex surfaces, inequalities involving the diameter, the area and the length of simple closed quasi-geodesics (jointed work with C. Vilcu).Using the above simple closed quasi-geodesics we proved that any polyhedra are unfolded to a planar simple polygon by some cutting (jointed work with J. O'Rourke, C. Vilcu).We discussed several other unfolding by using simple clodes quasi geodesics, also