2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature
2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature
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非正曲率中的 2 框架流动动力学和双曲秩刚度
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发表时间：
2007
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通讯作者：
D. Constantine
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作者：
D. Constantine
本文给出了秩为1的非正曲空间的双曲秩刚性结果。设M$是一个具有非正截面曲率的秩为1的紧致流形，并假定沿M$中的沿着每一测地线都有一个与测地线方向曲率为-a^2$的平行向量场.我们证明了如果M是奇维的，或者如果M是偶维的，并且截面曲率压缩为：-λ ^2> 0.93，则M的常曲率等于-a^2。当$-a^2$是曲率上界时，这给出了Hamenst”{a}dt的双曲秩刚性定理的一个较短的证明，在偶数维中服从拼挤条件;在所有其他情况下，这是一个新的结果。我们还提出了一个刚性的结果，只使用一个假设最大的李雅普诺夫指数直接类比康奈尔所做的工作。通过假设严格的负曲率，主要定理的证明被大大简化;事实上，在除了7维和8维之外的所有维中，它是$（dim（M）-1）$-标架流遍历性的直接结果。在这些特殊的尺寸，诉诸动力学的2-框架流必须作出和计划的证明，可以推广到处理秩1，非正弯曲的空间。
This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Lambda^2 >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity theorem of Hamenst"{a}dt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof of the main theorem is simplified considerably by assuming strict negative curvature; in fact, in all dimensions but 7 and 8 it is then an immediate consequence of ergodicity of the $(dim(M)-1)$-frame flow. In these exceptional dimensions, recourse to the dynamics of the 2-frame flow must be made and the scheme of proof developed there can be generalized to deal with rank 1, nonpositively curved spaces.