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In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions : on a smooth manifold ''M'', their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. == An example == Marston Morse proved that, provided is compact, any smooth function : could be approximated by a Morse function. So for many purposes, one can replace arbitrary functions on by Morse functions. As a next step, one could ask, 'if you have a 1-parameter family of functions which start and end at Morse functions, can you assume the whole family is Morse?' In general the answer is no. Consider, for example, the family: : as a 1-parameter family of functions on . At time : it has no critical points, but at time : it is a Morse function with two critical points : Jean Cerf〔(French mathematician, born 1928 )〕 showed that a 1-parameter family of functions between two Morse functions could be approximated by one that is Morse at all but finitely many degenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when an index 0 and index 1 critical point are created (as increases). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cerf theory」の詳細全文を読む スポンサード リンク
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