- 作者: 熊璩
- 作者服務機構: 國立臺灣大學數學系
- 中文摘要: 自從 J. Mather 在研究可微函數奇異點的理論有了重大的突破以後(請參考他那著名的七篇論文,從1967到1973),在這個領域和許多相當領域;裏都有了相常大的發展。將 Mather 的想怯應用到微分形式的奇異點上的努力就該是很自然的了。在1970 年 J. Martinet 開始提出來了這個問題,其後有好幾個人都對這個問題發生了興趣。Martinef 的文章和作者的學位論文裏都是從局部的觀點來處理微分形式穩定性的問題,而且頗穫成功。這一篇論文的目的就是想把這個想法用在大域微分式的考慮,我們發現這種看起來極為自然的推廣是不太有趣的。我們證明了在一個緊緻流形M上並沒有Ck - 穩定(k≦∞)的P-微分形式存在,也沒有Ck- 微穩定(k<∞)的 P-微分形式存在,此處 1≦p≦dim M.對微穩定性我們也給了一個局部p-微分形式的一個否定性的結果(對 dim M≧4).
- 英文摘要: Ever since J. Mather made the break-through in the study of singul-arities of differential mappings (as can be found in his seven celebratedpapers ranging from 1967 to 1973), much progress in this field and manyrelated fields have been made. It is a natural step to try to applyMather's idea to study the singularities of differential forms. There aremany people working in this field after J. Martinet raised the problemin 1970. Both J. Martinet's paper & the author's thesis attack this problemfrom a local point of view and have proven to be very successful. It isvery tempting, so, to extend the idea of stability to a global version aswell. The attempt of this work is to study the stability of globally defineddifferential forms on compact manifold M. We show that the globalproblem, under the "natural" definition, is not a very exciting one byproving that there is no Ck-stable p-forme (k≦∞) and no Ck-infinitesim-ally stable p-forms (k<∞), where l≦p≦dim M. We also give a negativeresult on Ck-infinitesimal stability (k≦∞)for local 2-forms for dim M≧4,2≦p≦n-2.
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