Closed Geodesics And L-S Category Of Compact Lie Groups

This is a graduation thesis on the closed geodesic and L-S category of compact Lie groups.It mainly focuses on two problems:（1）The properties of the closed geodesic set of compact Lie groups;（2）Calculation of L-S category of compact Lie groups.Firstly,some properties of its closed geodesic set are described according to the Cartan theorem of Lie groups,and the relationship between the closed geodesic set of compact Lie groups and the closed geodesic set of its Cartan subgroups is given.The main results are S=∪g∈Gg-1S-1g.Secondly,according to the estimation inequality of L-S category of fiber bundle cat（E）≤catE（F）.cat（B）and the L-S category estimation inequality of compact Lie groups cat（G）≤1/2（dim（G）-rank（G））,we get what the L-S category of SO（n）is.The main result is cat（SO（n））=cuplength（SO（n））+1 when n≤9.Thirdly,according to the root system of Lie algebra,we describe the covering of compact Lie groups and give specific coverage by taking SU（2）as an example.Thus,the upper bound estimation of the number of L-S category of compact Lie groups is obtained,the main conclusion is cat（G）≤∑Ti=∑i=1（（?））.From the above results,we construct the（?）of SU（n）and give specific coverage of SU（n）.So we can get cat（SU（n））=n and cat（U（n）=n+1）directly.Finally the relationship between the problems we consider and the Ganea conjecture:cat（X×Sn）=cat（X）+1 for any finite complex X and the calculation of the L-S category of other manifolds are described.