A local Classification of Morse Funnction = Morse함수의 국소적인 분류
저자
Jeong, Gwon-Soo (Department of Mathematics, Mokpo National College) ; Kim, Yeong-Cheol ; Song, Yeong-Moo
발행기관
학술지명
권호사항
발행연도
1985
작성언어
English
KDC
040.000
자료형태
학술저널
수록면
351-356(6쪽)
제공처
소장기관
본 논문에서는, smooth한 좌표 변환을 이용하여, 파국 이론에 필요한 Morse함수를 국소적으로 분류한다.
Smooth함수 F:??x??→R가 (x,c)=0에서의 Hessian행렬 Hf?? ??이 nondegenerate이면, '함수 F는, smooth한 좌표 변환을 하여, ±?? 형태의 함수들과 국소적으로 동등하게 된다는 것을 보였다.
Let f:??→R be a smooth function. A point u∈?? is a critical point of f if Df??=0 or in coordinates ??=0. The value f(u) at a critical point u is called a critical value of f. We say that f has a nondegenrate critical point at u if Df??=0 and if ?? is a nondegenerate quadratic form(That is, its rank is equal to the number of variables n). Equivalently, the Hessian
Hf??
is non-singular, so the Hessian determinant det (Hf??
We say that f is Morese if the critical point at 0 is nondegenerate, and that f is structurally stable if for all sufficiently small smooth functions p, the critical points of f and f+p have the same type.
Proposition 1.
Let f:??→R be smooth in a neighborhood of o, with f(0)=0. Then in a possibly smaller neighborhood, there exist functions ??→R such that
??
where each ?? is smooth, and ??(0)=??
Proof.
We have f(??
=??
=??
Hence we define
??
Differentiation partially with respect to ?? show that ??
Lemma 2. (Morse Lemma)
Let u be a nondegenerate critical point of the smooth function f:??→R. Then there is a local coordinate system(??0 in a neighborhood U of u, with ??(u)=0 for all i, such that f=f(u)-?? for all y∈U.
Proof.
We can translate the origin to u, and hence assume u=0, and f(u)=f(0)=0.
By Proposition 1, we may then write f(x)= ??
in some neighborhood of 0. Since 0 is a critical point we have ??
Hence, using Proposition 1 again, there exist smooth functions ?? such that
??
and we can write
(*) f(x)=??
If we replace ?? by
?/
then ??=?? and
f(x)=??
Partially differentiating(*) twice, we see that
??
and hence the matrix
??
is non-singular since 0 is a nondegenerate critical point. Suppose inductively that there exist local coordinates ??…,?? in a neighborhood ?? of 0 such that
f=??
where ??=??. By a linear change in the last n-r+1 coordinates, we may assume that
?? Let
g(??
by the Inverse Function Theorem this is smooth in some neighborhood ?? of 0, contained in ?/
We change coordinates to ?? defined by
??
??
which, again using the Inverse Function Theorem, is a local diffeomorphism within some sufficiently small neighborhood ?? of 0, contained in ??. Now
??
a formula like that for the ??, but with r replaced by r+1. Hence, by induction, we obtain the conclusions of the Theorem.///
A function of the form
??
is called a morse 1-saddle. Hence the Lemma 2 implies that every nondegenerate critical point may be transformed by a diffeomorphism(smooth reversible coordinate change) to a Morse
1-saddle, for some 1.
Two smooth function f, g:??→R are said to be equivalent around 0 if ther is a local diffeomorphism y:??→?? around 0 and a constant γsuch that, around 0,
g(x)=f(y(x) + γ.
Then y is a smooth reversible local change of coordinates, and the shear thermγ adjusts the value of the function at 0, taking care of the various translation, of the origin used above.
Then Lemma 2 say that near a nondegenerated critical point a function is equivalent to one of the Morse standard forms.
Theorem 3.
Let f:??→R be a smooth function, with Df(0)=0 whose Hessian at 0 has rank r (and corank n-r). Then f is equivalent, around 0, to a function of the form
??
where f:??→R is smooth.
Proof.
By a linear change of coordinates u=u(x), we can transform the Hessian of f at 0 into the form
??
The Implicit Function theorem now allows us to express the set
??
as the graph
??
of a smooth function
g:??
We use g to turn this graph into the (??)-axis, by a map ??, easily seen to be a local diffeomorphism, defined by
??
Let F=f??. Locally, each function
F(??):??→R (??)→F(??)
has a mose critical point at the origin of ??, though not necessarily taking the value 0 at that point. We write
??
Now the argument appearing in the first part of the proof of the Morse Lemma may be used(after generalizing Proposition 1 to the case where f vanishes along a multi-axis) to write
F(u)=??
where for each choice of ??
??:??→R
is smooth. The remainder of the proof of the Morse Lemma, applied in a (??)-dependent fashion to this expression, reduces F to the form
??
and prove the 소대그. In essence, the whole process of reduction to standard form used to prove the Morse Lemma depends smoothly on ??once the initial tidying has been done.
Theorem 4.
Let F:??x??→R be smooth. Denote a point in ??x?? by(x,c)=(??).
Suppose that the Hessian
H=??
has corankmat (x,c)=0. Then F is equivalent to a family of the form
??
Proof.
Find a nondegenerate (n-m) x (N-m) minor of H and renumber the coordinates to make it
??
Now define
G:?? x ?? →??
(x,c)→??
This is of maximal rank N-m on ??x??, by hypothesis, Hence by the Implicit Function
쏘대그 there is locally a function(defined on a neighborhood U of 0 in ??x??)
g:??x??→??
such that
g(g(??), ??=0
g(0)=0
By continuity, if we put
z=(g(??), ??)
the matrix
??
is nondegenerate for all c∈U. Define
??
??
This is clearly a diffeomorphism which preserves the sets c=constant. Let F=f??. Then
around 0,
??(0.…??)=0, for 1??,
??
Now we may proceed by induction as in the proof of the Lemma 2(Morse Lemma) except that the ??, ??, g etc. (for 1??i, j??N-m) have (??) as parameters. For fixed parameter values we only change ?? in the process. Since the critical value of each
F??
is unchanged, and defines the function F:??x ??→R in the statement of the theorem, we obtain the conclusion of the theorem.///
Theorem 5.
Let f;??x??→R be smooth. Suppose that the Hessian ?? is nondegenerate at (x,c)=0. Then F is equivalent to family of the form
??
Proof.
Put m=0 in the theorem 4, then as corollary, it is obvious.///
For a family of functions expressed in the form given in Theorem, we know that ?? are the essential variables, and ?? the inessential variables. This reflects the fact that for many purpose we can neglect the effect of ??
Theorem 5 gives us the stability of More function in a particularly strong form. The function ??(y)=F(y,c) is Morse, of fixed type, and independent of c. And any small perturbation of a function f:??→R with a Morse singularity at 0∈?? can be vainshed around 0∈?? by a reparametrization of ?? and the addition of a constant, restoring the original form.
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