Fixed Points and Torsion on Kahler Manifolds
Fixed Points and Torsion on Kahler Manifolds
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DOI：
10.2307/1969889
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发表时间：
1959-07
影响因子：
4.9
通讯作者：
T. Frankel
中科院分区：
文献类型：
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作者：
T. Frankel
当一个单参数群等距作用在黎曼流形M上时，不动点集F表现良好。已知F的每个分量Fo 0是M的全测地子流形，其维数与M的维数具有相同的奇偶性（例如，参见，S.小林，等距不动点，名古屋数学杂志，13（1958），63-68）。当M是紧的Kihler时，等距是全纯变换，FM是紧的Kihler子流形（可以化为点）;特别地，作为圈，这些分量不能在M中有界。本文主要研究Kihlerian情形下不动点集的结构;然而，复结构的使用主要是为了方便;我们的结果也适用于基本外2-形式是调和的特殊类型的辛流形。正如有几个人已经向我们指出的那样，我们的情形等价于有一个对紧致Kahler M作复解析作用的toral群。Bott [2]给出了某些齐性空间和loop空间与群的同调的一些重要结果。我们的主要结果，定理和推论？4可以被认为是前者的直接推广（见推论3）。我们的方法产量，在同一时间，新的证明他的结果。我们的证明是简单的应用程序的另一个阶段博特的工作，即他的扩展的莫尔斯理论的临界点的职能与“非退化的临界流形”[3]。下面的简单示例说明了该方法。设S2是2-球面，设'1 t是S2绕z轴旋转的单参数群。（D）的固定（或静止）集合F由北极和南极组成，即，速度矢量X消失的地方。现在
When a 1-parameter group acts by isometries on a Riemannian manifold M, the fixed point set F is nicely behaved. It is known that each component Fo0 of F is a totally geodesic submanifold of M whose dimension has the same parity as the dimension of M (see e.g., S. Kobayashi, Fixed points of isometries, Nagoya Math J., 13 (1958), 63-68). When M is compact Kihler theisometries are holomorphic transformations and the FM are compact Kihler submanifolds (which may reduce to points); in particular, as cycles, these components cannot bound in M. This paper is mainly concerned with the structure of the fixed point set in this Kihlerian case; however, the use of the complex structure is mainly for convenience; our results also hold for the special type of symplectic manifold in which the fundamental exterior 2-form is harmonic. As has been pointed out to us by several people, our situation is equivalent to having a toral group acting complex analytically on a compact Kahler M. Bott [2] has given some important results on the homology of certain homogeneous spaces and the loop space to a group. Our main results, the theorem and corollaries of ? 4 can be considered as direct generalizations of the former (see Corollary 3). Our method yields, at the same time, new proofs of his results. Our proofs are simple applications of another phase of Bott's work, namely his extension of the Morse theory of critical points to functions with "non-degenerate critical manifolds" [3]. The following simple example illustrates the method. Let S2 be the 2-sphere and let '1t be the 1-parameter group of rotations of S2 about the z axis. The fixed (or stationary) set F of (D, consists of the north and south poles, i.e., the places where the velocity vector X vanishes. Now