Morse theory for $G$-manifolds
Morse theory for $G$-manifolds
复制标题
$G$流形的莫尔斯理论
DOI：
10.1090/s0002-9904-1965-11306-4
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发表时间：
1965
影响因子：
1.3
通讯作者：
A. Wasserman
中科院分区：
文献类型：
--
作者：
A. Wasserman
Morse理论将Hubert流形的拓扑[3，§9]，M，与只有非退化临界点的C函数/:M - >R的行为联系起来。在将莫尔斯理论应用于研究G流形，即紧李群G作为可微变换群的流形时，当然必须使用范畴中的映射，即等变映射。然而，如果x是一个等变函数的临界点，那么gx也是任意g的临界点，因此必须允许临界轨道，或者更一般地说，允许临界子流形。在§1中我们给出了必要的定义和符号。在§2中，我们推广了R. Palais在[3]中的结果，研究了完全黎曼g空间ikf上的不变C函数f: M-+R，其中除了满足条件(C)[3，§10]外，我们还要求f的临界轨迹是Bott意义上的非退化临界流形的并。在§3中，我们证明了如果M是有限维的，那么M上的任何不变C函数都可以用一个临界轨道是非简并的C不变函数来C逼近。与§2的结果一起，为小柄体分解技术的g流形提供了一个模拟。证明将在其他地方给出。
Morse theory relates the topology of a Hubert manifold [3, §9], M, to the behavior of a C function/: M—>R having only nondegenerate critical points. In applying Morse theory to the study of G-manifolds, i.e., manifolds with a compact Lie group G acting as a differentiable transformation group, one must, of course, use maps in the category, i.e., equivariant maps. However, if x is a critical point of an equivariant function then gx is also a critical point for any g £ G , hence one must allow critical orbits or, more generally, critical submanifolds. In §1 we give the necessary definitions and notation. In §2 we extend the results of R. Palais in [3 ] to study an invariant C function ƒ : M-+R on a complete Riemannian G-space ikf, where in addition to ƒ satisfying condition (C) [3, §10], we require that the critical locus of ƒ be a union of nondegenerate critical manifolds in the sense of Bott [ l ] . In §3 we show that if M is finite-dimensional then any invariant C function on M can be C approximated by a C invariant function whose critical orbits are nondegenerate. Together with the results of §2 this provides an analogue for G-manifolds of the Smale handlebody decomposition technique. Proofs will be given elsewhere.