The higher rank rigidity theorem for manifolds with no focal points
The higher rank rigidity theorem for manifolds with no focal points
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DOI：
10.1007/s10711-012-9776-3
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发表时间：
2011-11
影响因子：
0.5
通讯作者：
Jordan Watkins
中科院分区：
文献类型：
--
作者：
Jordan Watkins
我们称一个黎曼流形M的秩M ≥ k，如果M的每一条测地线至少有k个平行Jacobi域。Ballmann和Burns-Spatzier的秩刚性定理，后来被Eberlein-Heber推广，指出一个完备的，不可约的，单连通的黎曼流形M，秩k ≥ 2（“高阶”假设），其等距群Γ满足条件，即在SM中的Γ-递归向量是稠密的，是一个非紧型对称空间。例如，这包括允许有限体积商的较高秩M。我们采用Ballmann和Eberlein-Heber的方法证明了这一定理的推广，其中流形Mis假设只有没有焦点。然后，我们使用这个定理推广到没有焦点的结果Ballmann-Eberlein指出，紧凑的流形的非正曲率，秩是一个不变量的基本群体。
We say that a Riemannian manifoldMhas rank M ≥kif every geodesic inMadmits at leastkparallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifoldMof rankk≥ 2 (the “higher rank” assumption) whose isometry groupΓsatisfies the condition that theΓ-recurrent vectors are dense inSMis a symmetric space of noncompact type. This includes, for example, higher rankMwhich admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifoldMis assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.