Uniform Morse lemma and isotopy criterion for Morse functions on surfaces |
| |
Authors: | E A Kudryavtseva |
| |
Institution: | 1.Faculty of Mechanics and Mathematics,Moscow State University,Leninskie Gory, Moscow,Russia |
| |
Abstract: | Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $
\mathcal{D}_0
$
\mathcal{D}_0
⊂ Diff(M) be the group of diffeomorphisms homotopic to id
M
. Two smooth functions f, g: M → ℝ are called isotopic if f = h
2 ℴ g ℴ h
1 for some diffeomorphisms h
1 ∈ $
\mathcal{D}_0
$
\mathcal{D}_0
and h
2 ∈ Diff+(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions
from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which
continuously and Diff(M)-equivariantly depends on f in C
∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|