A class of Riemannian metrics on a manifold
A class of Riemannian metrics on a manifold
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流形上的一类黎曼度量
DOI：
10.4310/jdg/1214428438
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发表时间：
1968
影响因子：
2.5
通讯作者：
Hideki. Omori
中科院分区：
文献类型：
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作者：
Hideki. Omori
R. Bott 在他的启发性论文 [3] 中证明，如果从黎曼流形 M 中的点 p 开始的大地测量都是闭合大地测量，其一圈长度是恒定的，那么在计算重数时，这些闭合大地测量的一圈上 p 的共轭点的数量是恒定的。 Nakakawa [9] 最近扩展了这一结果，他证明，如果所有从点 p 开始且长度 c 恒定的测地线都回到点 p（这些测地线不一定是闭合测地线），则计算重数时，这些闭合测地线段的一圈上的共轭点的数量是恒定的。如果假设更强的条件，使得 p 相对于从 p 开始的每条测地线的割点可能成为该闭合测地线段的中点，则流形 M 具有分解 M = Dp \j φDN，如 Warner 的论文 [11] 所示，其中 Dp 是圆盘，N 是 p9 的割轨迹，在这种情况下成为闭合子流形，DN 是 M 中 N 的正常圆盘丛。这些事实证明，如果紧连通实解析黎曼流形 M 具有子流形 N，使得 N 相对于从 N 开始且初始方向与 N 正交的每条测地线的割点与 N 的距离为常数 π，则 M 具有分解 M = DN (J φDN,，其中 N' 是 N 的割轨迹，DN、DN 分别是 N、N' 的法向盘丛（参见定理 3.1）。当然，具有这种分解的流形是非常特殊的，但无论如何，在单个流形 M 上，有许多不同的黎曼度量，它们形成了一个凸集，但是，每个黎曼度量都应该受到流形的拓扑结构的影响，粗略地说，人们必须能够仅使用一个黎曼度量来确定 M 的拓扑结构。度量，但至少目前看来是不可能的，因此，考虑一些有用的黎曼度量而不是单个度量或整个度量似乎很有趣。
In his suggestive paper [3], R. Bott proved that if geodesies starting from a point p in a riemannian manifold M are all closed geodesies whose length of a lap is constant, then the number of conjugate points of p on a lap of these closed geodesies are constant, counting the multiplicity. This result has been extended recently by Nakagawa [9], who proved that if all geodesies starting from a point p with a constant length c come back to the point p (these are not necessarily closed geodesies), then the number of conjugate points on a lap of these closed geodesic segments are constant, counting the multiplicity. If a stronger condition is assumed so that the cut point of p with respect to every geodesic starting from p may become a middle point of this closed geodesic segment, then the manifold M has a decomposition M = Dp \j φDN, as it is seen in Warner's paper [11], where Dp is a disk, N is a cut locus of p9 which becomes a closed submanifold in this case, and DN is a normal disk bundle of N in M. In this paper, as an extension of these facts, it will be proved that if a compact connected real analytic riemannian manifold M has a submanifold N such that the cut point of N with respect to every geodesic, which starts from N and whose initial direction is orthogonal to N has a constant distance π from N, then M has a decomposion M = DN (J φDN,, where N' is the cut locus of N and DN, DN, are normal disk bundles of N, N' respectively (cf. Theorem 3.1). Of course, manifolds having such a decomposition are very special, but at any rate, it seems interesting to consider some details about that kind of manifold. On a single manifold M, there are many, various riemannian metrics, which form a convex set. Each of these riemannian metrics, however, ought to be influenced by the topological structures of the manifold. Roughly speaking, one must be able to determine the topological structures of M by using only one riemannian metric, but at least at the present time it seems impossible. Therefore, it seems interesting to consider some useful class of riemannian metrics instead of a single metric or the whole metrics. In this paper, it