The marked length-spectrum of a surface of nonpositive curvature☆
The marked length-spectrum of a surface of nonpositive curvature☆
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DOI：
10.1016/0040-9383(92)90013-8
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发表时间：
1992-10
期刊：
影响因子：
--
通讯作者：
C. Croke;A. Fathi;J. Feldman
中科院分区：
文献类型：
--
作者：
C. Croke;A. Fathi;J. Feldman
1.背景我们在M上确定了一个(严格)负曲率的参考黎曼度量Go。下面的定理是由Morse[Lo].定理1.1(Morse)得到的。设g是M上的黎曼度量，g和Go是g的Zifts和G的黎曼度量Go的(严格)负曲率到M的泛覆盖Fi，则存在一个常数K>；0，它只依赖于g和Go，使得任何Go-测地线在其K-邻域中包含极小g-测地线，并且任何极小g-测地线在其K-邻域中包含唯一的Go-测地线。图@：2-+从&极小化测地线空间2到Go-测地线空间Go是连续且适当的(当g没有共轭点时，则&？当然是所有g-测地线的空间。)此外，地图。！？是n，(M)等变。
1. BACKGROUNDWe fix a reference Riemannian metric go of (strictly) negative curvature on M. The following theorem is due to Morse [lo].THEOREM 1.1.(Morse). Let g be a Riemannian metric on M. Let g and go be the Zifts of g and of the Riemannian metric go of (strictly) negative curvature to the universal cover fi of M. Then there exists a constant K> 0, which depends only on g and go, such that any go-geodesic contains in its K-neighborhood a minimizing g-geodesic, and any minimizing g-geodesic contains a unique go-geodesic in its K-neighborhood. The map@: 2-+ go from the space 2 of &minimizing geodesics to the space go of go-geodesics which sends a minimizing g-geodesic to the go-geodesic in its K-neighborhood is continuous and proper.(When g has no conjugate points then &? is of course the space@ of all g-geodesics.) Moreover, the map.!? is n,(M) equivariant.