Deformation of Structures on Manifolds Defined by Transitive, Continuous Pseudogroups Part I: Infinitesimal Deformations of Structure
Deformation of Structures on Manifolds Defined by Transitive, Continuous Pseudogroups Part I: Infinitesimal Deformations of Structure
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由传递连续伪群定义的流形上的结构变形第一部分：结构的无穷小变形
DOI：
10.2307/1970277
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发表时间：
1962
影响因子：
4.9
通讯作者：
D. Spencer
中科院分区：
文献类型：
--
作者：
D. Spencer
现在我们从本文第一部分所关心的F-结构的无穷小变形转到局部r-结构的变形，或者说转到局部r-结构的变形的芽，因为我们只对小变形感兴趣。无穷小变形的层，如层JA，具有分次李代数的结构;变形芽的层，在对参数的依赖是可微的情况下，具有群的结构。设r是一个传递的连续伪群，M是n维r-流形.在开始的时候？2定义了M上群的层g_（9 ms），即P的局部变换芽的层g_（9 ms）可微地依赖于m个参数t ′，t ~ 2，l。* ，tin，其中（t '，t2，. **，tin）是Rm的原点0附近的点t。我们现在为每个非负整数It引入M上的群的层Jmp），其中Jc~m）是局部可微变形的芽的层，即，局部可微纤维保持变换，主丛PA在M上，可微地依赖于m个参数。更准确地说，令2 q ' = P> x RM，CV = Mx R”，并将PA M与位于R“的原点0上的7 r：TAs RM的纤维识别。特别地，M被识别为R”的点0（原点）上的-r：CV）Rm的纤维，即，M= t-1（0）（比较？2）。
We now turn from the infinitesimal deformations of F-structure, which are the concern of Part I of this paper, to the deformations of local rstructure, or rather to the germs of deformations of local r-structure, since we are interested only in small deformations. The sheaves of infinitesimal deformations, such as the sheaves JA, have structures of graded Lie algebras; the sheaves of germs of deformations, in the cases where the dependence on the parameters is differentiable, have structures of groups. Suppose that r is a transitive, continuous pseudogroup, M a r-manifold of dimension n. At the beginning of ? 2 we defined the sheaf g9ms of groups over M, namely the sheaf 9() of germs of local transformations of P depending differentiably on m parameters t', t2,l. *, tin, where (t', t2,. **, tin) is a point t in the neighborhood of the origin 0 of Rm. We now introduce, for each non-negative integer It, a sheaf JMP) of groups over M, where JC~m) is the sheaf of germs of local differentiable deformations, i.e., local differentiable fibre-preserving transformations, of the principal bundle PA over M, which depend differentiably on m parameters. More precisely, let 2q' = P> x RM, CV = Mx R", and identify PA M with the fibre of 7r: TAs RM which lies over the origin 0 of R?. In particular, M is identified with the fibre of -r: CV ) Rm over the point 0 (origin) of R", i.e., M= t-1(0) (compare ? 2).