Metric with ergodic geodesic flow is completely determined by unparameterized geodesics
Metric with ergodic geodesic flow is completely determined by unparameterized geodesics
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遍历测地线流的度量完全由非参数化测地线确定
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
P. Topalov
中科院分区：
文献类型：
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作者：
V. Matveev;P. Topalov
设g是具有遍历测地线流的黎曼度量。然后，如果某个度量G与g具有相同的测地线（视为非参数化曲线），则这些度量是位似的。如果亏格大于1的闭曲面上的两个度量具有相同的测地线，则它们是位似的。设g，n是n ≥ 2维流形M上的两个C-光滑黎曼度量.如果它们具有相同的测地线，则它们是射影等价的。对于给定的度量g，总是存在射影等价度量的平凡例子：对于任何正常数C，度量Cg射影等价于g。考虑由G = g <$αj给出的纤维线性映射G：TM → TM和由A def =（det（G））1 n+1G−1给出的纤维线性映射A：TM → TM。考虑特征多项式det（A − λ Id）= c 0 λ + c1λ + · · · + cn和映射S 0，S1，. . .，Sn−1：TM → TM，由下式给出：
Let g be a Riemannian metric with ergodic geodesic flow. Then if some metric ḡ has the same geodesics (regarded as unparameterized curves) with g, then the metrics are homothetic. If two metrics on a closed surface of genus greater than one have the same geodesics, then they are homothetic. Let g, ḡ be two C-smooth Riemannian metrics on a manifold M of dimension n ≥ 2. They are projectively equivalent if they have the same geodesics regarded as unparameterized curves. For a given metric g, there always exist trivial examples of projectively equivalent metrics: for any positive constant C, the metric Cg is projectively equivalent to g. Consider the fiberwise-linear mapping G : TM → TM given by G = gḡαj and the fiberwise-linear mapping A : TM → TM given by A def = (det(G)) 1 n+1G−1. Consider the characteristic polynomial det(A − λ Id) = c0λ + c1λ + · · · + cn and the mappings S0, S1, . . . , Sn−1 : TM → TM given by