奇点理论视角下伪球面上类光子流形的几何性质研究

In physics, there are some important physics meaning about lightlike curves, lightlike surfaces, lightlike hypersurfaces and other lightlike submanifolds. Physicists pointed out that the black hole horizon is kind of lightlike hypersurfaces, and lightcone and wave front of lightwave is a kind of lightlike hypersurface. Therefore, by the studying of lightlike manifolds, we can obtian some intrinsic geometric properties of the black hole horizon. By Lagrangian/legendrian singularity theory, this project will focuse on explaining the singularities and topology of lightlike curves, lightlike surfaces and lightlike hypersurfaces on pseudo-spheres with index two. This project mainly includes the following aspects: ①. Giving geometry properties of Bertrand curves and AW(k)-type curves of lightlike curves on three dimention pseudo spheres, looking for new geometric invariants of light curve, giving the relations between the geometric invariants and the singularities of those lightlike curves by the contact with spheres; ②. Studying the singularities of the Gauss indicatrices of lightlike surfaces on three dimensional pseudo spheres, the singularity properties of asymptotic differential equations and principal curvature differential equations of asymptotic line and principal normal line on lightlike surfaces are studied; ③. Studying the singularities and geometry properties of new hypersurfaces, which generated by lightlike hypersurfaces on n dimensional lightcone.

在物理学中，类光曲线，类光曲面和类光超曲面等类光子流形有着重要的物理学意义。物理学家指出：黑洞视界实际上就是类光超曲面，同时光锥面和光波的波前面都是类光超曲面。因此，通过类光子流形的几何性质研究能够反映黑洞视界的一些内蕴性质。本项目主要以Lagrangian和legendrian奇点理论为研究工具，研究指标数为2的伪球面上类光曲线，类光曲面和类光超曲面的奇点和拓扑性质。本项目主要包括以下几个面：①. 研究3维伪球面上类光曲线的Bertrand曲线和AW(k)-型曲线的几何性质，寻找类光曲线新的几何不变量，通过曲线与某些球面的切触研究类光曲线的奇点类型与几何不变量的内在联系；②. 研究3维伪球面上类光曲面高斯指标族的奇点类型，同时对类光曲面上渐近线与主法线所对应的渐近微分方程和主法曲率微分方程的奇异性进行研究；③. 研究n维光锥面上类光超曲面沿着类光方向所生成的新超曲面的几何性质和奇点。

黑洞视界内的物质都不可避免地要坍缩到黑洞中心，成为一个中心奇点（宇宙奇点），同时物理学家指出黑洞视界实际上就是Minkowski空间中类光超曲面，同时光锥面和光波的波前面都是类光超曲面，这给黑洞视界赋予了几何意义。因此对伪欧氏空间中类光超曲面的几何性质的研究具有物理意义。本项目主要从几何视角来研究Minkowski空间中类光超曲面的奇点来反映宇宙奇点局部上的类型及性质。本项目主要研究了以下几个方面：①. 研究3维球面上磁曲线的Bertrand曲线和AW(k)-型曲线的几何性质，得到磁曲线新的几何不变量，得到磁曲线的内蕴性质；②. 研究3维伪球面上类光磁曲现的奇点类型，同时对由类光磁曲线所得到的的类光曲面上渐近线与主法线所对应的渐近微分方程的奇异性进行研究；③. 研究n维光锥面上伪类光超曲面奇点分类和几何性质。