On Manifolds All of Whose Geodesics are Closed
On Manifolds All of Whose Geodesics are Closed
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关于所有测地线均闭合的流形
DOI：
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发表时间：
1954
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影响因子：
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通讯作者：
R. Bott
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文献类型：
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作者：
R. Bott
黎曼流形 M 上长度为 w 的闭测地线 g 表示实数 [oc 0 的映射 g: R -> M。如果弧 g(x) (O 0，则 M 是单连通的，则 g 被称为简单的，并且其积分上同调环是截断多项式环，由维度 X + 1 的元素 0 生成。截断多项式环是从 Z(x) 获得的环，Z(x) 是整数，通过添加单个关系 xn = 0。通过该定理对 H(M) 施加的限制非常强：X + 1 必须整除 dim M；如果 X + 1 是奇数，则 X + 1 = dim M；M 在任何域 K 上的庞加莱多项式的形式为 P(t) = 1 + t+1 + t2(X+) ... tk(X+)。例如，J. Adem 最近获得了复数的上同调环多项式环，从他的工作中可以看出，只要 02 5z 0，并且如果 dim 0 > 8，则 O' = 0。其上同调环属于所讨论的类型的唯一已知的简单连通流形是：A. 球体 S'；B. 复射影空间； Zn; (n _ 1); C. 四元数射影空间 Qf; (n > 1); D. 凯莱射影平面 C2 。第 375 章
By a closed geodesic g of length w on a Riemann manifold M is meant a map g: R -> M of the reals [oc 0. g is called simple if the arc g(x) (O 0, then M is simply connected, and its integral cohomology ring is a truncated polynomial ring, generated by an element 0 of dimension X + 1. A truncated polynomial ring is a ring obtained from Z(x), the ring of polynomials over the integers, by adding the single relation xn = 0. The restrictions imposed on H(M) by virtue of this theorem are quite strong. For instance: X + 1 must divide dim M; if X + 1 is odd, then X + 1 = dim M; the Poincare polynomial of M over any field K is of the form P(t) = 1 + t+1 + t2(X+) ... tk(X+). Much more subtle restrictions on a truncated polynomial ring which is the cohomology ring of a complex have recently been obtained by J. Adem [5]. For instance, it follows from his work that dim 0 is a power of two, whenever 02 5z 0, and if dim 0 > 8, then O' = 0. The only known simply connected manifolds whose cohomology ring is of the type under discussion are the following ones: A. The spheres S'; (n _ 2); B. The complex projective spaces Zn; (n _ 1); C. The quaternion projective spaces Qf; (n > 1); D. The Cayley projective plane C2 . These spaces are precisely the irreducible symmetric spaces of rank 1 of Cartan [4]. Conversely, Cartan's classification shows that amongst symmetric spaces the irreducible ones of rank 1 are the only ones which have their geodesics simply closed at some point. Cartan's work therefore gives a much stronger result than our theorem under the strong additional condition that M be symmetric. 375