Morse-Sard type results in sub-Riemannian geometry
Morse-Sard type results in sub-Riemannian geometry
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Morse-Sard 类型导致亚黎曼几何
DOI：
10.1007/s00208-004-0622-2
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发表时间：
2005
影响因子：
1.4
通讯作者：
E. Trélat
中科院分区：
文献类型：
--
作者：
L. Rifford;E. Trélat
设（M，Δ，g）是一个次黎曼流形，x0 ∈ M.假设Chow条件成立，且M具有次黎曼距离是完备的，我们证明了存在M的稠密子集N1，使得对于N1的每个点x，都有一条唯一的极小化路径使x0指向x，这条轨迹允许一个正常的极值提升。若分布Δ处处为余秩1，则证明了M的满Lebesgue测度子集N2的存在性，使得对N2中的每一点x，都存在一条使x0指向x的极小化路径，该极小化路径允许正常极值提升，且非奇异，且点x不与x0共轭.特别地，次黎曼指数映射的像在M中是稠密的，并且在corank的情况下，一个映射在M中是满Lebesgue测度的。
Abstract.Let (M,Δ,g) be a sub-Riemannian manifold and x0 ∈ M. Assuming that Chow’s condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution Δ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M, and in the case of corank one is of full Lebesgue measure in M.