正曲率流形上对偶叶状结构的水平度量

本项目研究单位球面与正曲率流形上对偶叶状结构（简称DF）的水平度量。DF及其上的水平度量是由知名几何学家Wilking于07年提出的，它们源于奇异黎曼叶状结构（简称SRF）但又与次黎曼几何相关，这种交叉性使得它们受到关注。近几年将DF理论应用于SRF领域产生了许多重要成果，然而DF上的水平度量却没有得到任何发展。本项目致力于研究这片空白领域，并已取得一些有趣的成果，它们显示单位球面上的水平度量具有很强的刚性。通过运用比较几何的思想，我们将单位球面上的部分成果推广到了正曲率流形上。我们还发现在一些情形下水平度量可以反过来决定正曲率流形的黎曼度量。.本申请书解释了我们已取得的成果，并根据这些成果将我们进一步的研究目标分为三部分，相应地我们也提出了三个有趣的问题并给出了详尽的研究方案。

This project studies the horizontal metric of dual foliations(DF for short) on unit sphere and positively curved manifolds. DF and its horizontal metric were proposed by the famous geometer Wilking in 2007. They originated from singular Riemannian foliation(SRF for short) and also related to sub-Riemannian geometry. This intersectionality make them got attentions. In recent years, the application.of DF theory to the area of SRF has produced a lot of important achievements, however the horizontal metric of DF has never been studied. This project devotes to study this blank area, moreover we have got some interesting results which shows that the horizontal metric on the unit sphere has strong rigidity. By using the ideology of comparison geometry, we have generalized part of the results.on unit sphere to positively curved manifolds. We also found that sometimes the horizontal metric could determine the Riemannian metric of positively curved manifolds conversely..This application form explains our results, according to these results we divide our further research objects into three parts. We proposed three interesting questions accordingly, and also gave elaborate research approaches for them.