非负曲率亚历山大空间的结构

One of the most important classes of Alexnadrov spaces is noncompact nonnegatively curved Alexnadrov spaces. There is an open problem: is the Sharafutdinov retraction a submetry? Under the advision of my supervisor, in my Phd. thesis, we gave a positive answer for a special case to the open problem. On the basis of it, in this work, we want to study further some special noncompact, nonnegatively curved Alexandrov space. We want to study the following questions: (1) For topologically nice, noncompact nonnegatively curved Alexnadrov space with soul of codimension two, is the Sharafutdinov retraction a bundle map？(2) For such space under what condition can the space split? (3) For topologically regular, simply connected, noncompact nonnegatively curved Alexnadrov space with soul of codimension two, is the Sharafutdinov retraction a submetry？

亚历山大几何中很重要的一类研究对象是非负曲率, 非紧的亚历山大空间。对于这类空间有这样的公开问题：对非负曲率,非紧的亚历山大空间的Sharafutdinov形变收缩是不是一个submetry? 本人在博士期间，在导师的指导下对该问题的特殊情形给予了回答，在此基础上，本工作主要是围绕该问题对特殊的非负曲率, 非紧的亚历山大空间的结构做一些探讨。拟讨论以下问题: (1)对于Soul的余维数为2的非负曲率, 非紧的亚历山大拓扑比较好的（每点的每层方向空间都同胚于相应维数的拓扑球）空间Sharafutdinov 映射是不是一个纤维丛映射？(2)上述空间满足什么条件时是度量乘积空间？(3)对于余维数为2的非负曲率, 非紧的单连通的拓扑正则的（每点的方向空间都同胚于一个球）亚历山大空间Sharafutdinov 映射是不是submetry映射？

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