Research report: "The w - the germ of function defined on the manifold r-reticular" ppsx

ω-Determinancy of Function GermsNgo Dinh Quoca Abstract.. In this paper, we generalize theω-determinancy of a function germf ∈ εnunder the groupDif frnonr-reticular manifolds and give a

ω-Determinancy of Function Germs

Ngo Dinh Quoc(a)

Abstract In this paper, we generalize theω-determinancy of a function germf ∈ ε(n)under the groupDif fr(n)onr-reticular manifolds and give a necessary and sufficient condition that a function germf ∈ ε(n)is anω-determinancy under the groupDif fr(n) The main result is Theorem 3.1.

1 Introduction

In many years, the following question: “When is a smooth function f deter-mined in a neighbourhood of a point 0 by one of its Taylor polynomials at 0 in the sense that every another function having the same Taylor polynomial coincides with f around 0 up to a smooth change of coordinates?” has attracted considerable attention

In general an any function does not have this property at an arbitrary point; for instance, a flat function cannot be determined by its Taylor polynomial in the above sense But some noticeable results have been obtained in [2], [5], and [8] In

1968, John Mather [5] gave two conditions, one sufficient and the other necessary, for a function to be determined by its k-th Taylor polynomial In 1977, H H Vui,

N H Duc, N T Cuong, N T Dai, N S Minh [2] gave necessary and sufficient conditions that a function f ∈ ε(n) is ω-determined under the group Diff(n)

In 1980, H H Vui gave some results for map-germs to be ω- determined by its Taylor polynomial under the subgroup K|x 1 and <|x 1 of the group K and <, which consisting of all coordinate transformations preserving 0 and {x1 = 0}

The aim of this paper is to give a necessary and sufficient condition for a func-tion f ∈ ε(n) to be ω-determined under the group Diffr(n), and to generalize the

ω-determinancy of function-germs on a manifold with boundary, which is intro-duced by H H Vui in 1980

The paper contains two sections The first section deals with the basic no-tations In the section 2, we prove a necessary and sufficient condition that a function f ∈ ε(n) is ω-determined under the group Diffr(n) Our main result is Theorem 3.1

2 Preliminaries

We denote by ε(n, p) the set of germs of differentiable maps at zero from Rn

to Rp, and write ε(n) instead ε(n, 1) Fix r ∈ N, 0 ≤ r ≤ n Denote I = {1, , r},

1 - Received 30/09/2005, in revised form 20/01/2006.

and Xi is a germ of set {(x1, x2, , xn) ∈ Rn: xi = 0}for every i ∈ I Denote P (I)

by the family of all subsets of the set I The collection

X = {Xσ : Xσ =\

i∈σ

Xi, Xφ = IRn, σ ∈ P (I)}

is called a germ of a reticular manifold

Denote Noby the set of all positive natural numbers, and

Dif fr(n) = {Φ : (Rn, 0) → (Rn, 0) : Φ(Xσ) = Xσ, for all σ ∈ P (I)},

where Φ is a diffeomorphism germ at zero

2.1 Definition Let s be a nonnegative integer Two germs f and g ∈ ε(n) are said to be s-equivalentif Dγf (0) = Dγg(0) for all γ ∈ No with | γ |≤ s The class

of all germs s-equivalent to f is called the s-jet of f and is denoted by Js(f ) The set of all s-jets of germs of ε(n) is denoted by Js

n or simply by Js Js(f ) is called the convergence of its Taylor expansion

2.2 Definition Let s be a positive integer A germ f ∈ ε(n) is called s-determined under the groupDif fr(n)if for every germ g ∈ ε(n) satisfying Js(f ) = Js(g)there exists Φ ∈ Diffr(n)such that g = f ◦ Φ

2.3 Definition A germ f ∈ ε(n) is called ω-determined under the groupDif fr(n)

if for every germ g ∈ ε(n) satisfying the condition J∞(f )=J∞(g) there exists a

Φ ∈ Dif fr(n)such that g = f ◦ Φ

3 The main results 3.1 Theorem Letf : (Rn, 0) → (R, 0)be a smooth function germ fromRnintoR Then the following are equivalent

(a) f isω-determined under the groupDif fr(n);

(b) f isω-determined under the groupDif f (n)and the restrictionf0 =f |Xσ of

f onXσ isω-determined under the groupDif f (n)onε(n−| σ |), where| σ |

is the number of elements ofσ;

(c) m∞n ⊂< x1∂x∂f

1, , xr∂x∂f

r,∂x∂f

r+1, ,∂x∂f

n > ε(n);

(d) There existC, α, δ > 0such that

r

X

i=1

x2i(∂f

∂xi)

2

+

n

X

i=r+1

(∂f

∂xi)

2 ≥ Ck x kα, for allk x k< δ

Proof (a) ⇒ (b) Let f ∈ ε(n) Assume that f is ω-determined under the group Dif fr(n) We shall prove that for every g ∈ ε(n) satisfying J∞(f )= J∞(g), there exists Φ ∈ Diff(n) such that

Since f is ω-determined under the group Diffr(n), for every germ g ∈ ε(n) satisfying the condition J∞(f )=J∞(g) there exists a Φ ∈ Diffr(n) such that

g = f ◦ Φ Then Φ ∈ Diff(n), this implies that f is ω-deteminated under Dif f (n) It is remain to prove that the restriction f0 = f |Xσ is ω-determined under Diff(n) on ε(n− | σ |) Without loss of generality we can assume that

| σ |= k, k ≤ r Then we prove that f0 = f |x1= =xk=0 is ω-detemined under the group Diffr(n) on ε(n−k) Indeed, put x = (x1, , xk) , x0 = (xk+1, , xn), and f0(x0) = f0(xk+1, , xn) = f (0, , 0, xk+1, , xn) Let g ∈ ε(n−k) satisfying

J∞(f )(x0)= J∞(g)(x0) Then we have g(x0) = g(xk+1, , xn)and f0(x0) = f (0, x) Put g(x, x0) = f (x, x0) + ϕ(x0), where

ϕ(x0) = g(x0) − f0(x0) (3.2) Then we have g ∈ ε(n) Since g(xk+1, , xn) ∈ ε(n−k)such that J∞f0(x0) = J∞g(x0),

it implies that ϕ(x0) ∈ m∞n−k By (3.2) we have J∞(g) = J∞(f ) Thus by (a) there exists Φ(x, x0) ∈ Dif fr(n)such that f(x, x0) = g ◦ Φ(x, x0)and we have

f (0, x0) = g ◦ Φ(0, x0)

= f (Φ(0, x0)) + g(Φ(0, x0)) − f0(Φ(0, x0))

= g(Φ(0, x0))

Since f(Φ(0, x0)) = f0(Φ(0, x0)) and f(0, x0) = f0(x0), this follows that f0(x0) =

g ◦ Φ(0, x0) Hence f0(x0)is ω-determined under the group Diff(n) on ε(n−k) (b) ⇒ (c) Let (b) hold Since f is ω-determined under the group Diff(n) on ε(n)we have

m∞n ⊂< ∂f

∂x1, ,

∂f

and

f0 = f |x 1 = =xk=0 = f (0, x0)

is ω-determined under the group Diff(n) on ε(n−k), for every 0 ≤ k ≤ r Therefore

we get

m∞n−k ⊂< ∂(0, x

0)

∂xk+1 , ,

∂(0, x0)

∂xn > ε(n − k). (3.4) Put any ϕ(x, x0) ∈ m∞n, we have to prove that

ϕ(x, x0) ∈< x1

∂f

∂x1, , xr

∂f

∂xr,

∂f

∂xr+1, ,

∂f

∂xn > ε(n). (3.5) Since ϕ(x, x0) ∈ m∞n by (3.3) we have

ϕ(x, x0) =

k

X

i=1

∂f (x, x0)

∂xi ξi(x, x

0

) +

n

X

j=k+1

∂f (x, x0)

∂xj ηj(x, x

0

where ξi(x, x0) and ηj(x, x0) are germs in ε(n), which can be choosen such that

ξi(x, x0) ∈ m∞n , i = 1, 2, , k, and ηj(x, x0) ∈ m∞n , j = k + 1, , n

To prove (3.5), we firstly show that

m∞n ⊂< ∂f

∂xk+1, ,

∂f

∂xn > ε(n) + x1 xk.ε(n). (3.7) Ideed, let ξ(x, x0) ∈ m∞n Put

g(x, x0) = ξ(x, x0) − ξ(0, x0)

Obviously, we have g(x, x0) ∈ m∞n , and g(0, x0) = 0, then

g(x, x0) = x1x2 xkQ(x, x0) ∈ x1 xk.ε(n)

Hence

ξ(x, x0) = ξ(0, x0) + x1x2 xkQ(x, x0) (3.8) Because ξ(0, x0

) ∈ ε(n−k), then by (3.4) which follows that

ξ(0, x0) =

n

X

i=k+1

∂f (0, x0)

∂xi

but f(0, x0) ∈< f (x, x0) > ε(n), therefore

ξ(0, x0) ∈< ∂f

∂xk+1, ,

∂f

∂xn > ε(n) + x1 xk.ε(n). (3.10) From (3.8) and (3.10), we obtain ξ(x, x0) ∈< ∂x∂f

k+1, , ∂x∂f

n > ε(n) + x1 xk.ε(n) Therefore, (3.7) holds

Now we prove (3.5) Because ϕ(x, x0) ∈ m∞n, by (.37) the germs ξi(x, x0) in expression (3.6) are of the forms

ξi(x, x0) =

n

X

l=k+1

∂f

∂xlhl(x, x

0

) + x1 xk.h(x, x0), i = 1, 2, , k (3.11)

By replacing (3.11) into (3.6), we get

ϕ(x, x0) =

k

X

i=1

∂f

∂xi

h n

X

l=k+1

∂f

∂xlhl(x, x

0

) + x1 xk.h(x, x0)i+

n

X

j=k+1

∂f

∂xjξj

=

k

X

i=1

xi

∂f

∂xi

Y

j=1,j6=i

xj

 h(x, x0) +

n

X

j=k+1

∂f

∂xj

h

hj(x, x0)

k

X

i=1

∂f

∂xi + ξj

i This shows that

ϕ(x, x0) ∈< x1 ∂f

∂x1, , xr

∂f

∂xr,

∂f

∂xr+1, ,

∂f

∂xn > ε(n).

(c) ⇒ (d) Choose g(x1, , xn) = (x2

1+ +x2

n)2 Obviously g ∈ m4

n, thus g ∈ m∞

n From c) we have

g =

k

X

i=1

xi∂f

∂xiξi+

n

X

i=k+1

∂f

∂xiηi

≤

k

X

i=1

xi∂f

∂xi

2

+

n

X

i=k+1

∂f

∂xi

21 k

X

i=1

ξi2+

n

X

i=k+1

ηi2

1

≤ ch

k

X

i=1

xi∂f

∂xi

2

+

n

X

i=k+1

∂f

∂xi

2i12 ,

where c = suph

k

X

i=1

ξi2+

n

X

i=k+1

η2ii

1

, k x k≤ δ, with δ > 0 Thus we have

h k

X

i=1

xi∂f

∂xi

2

+

n

X

i=k+1

∂f

∂xi

2i12

≥ 1

c(x

2

1+ + x2n)2

k

X

i=1

xi∂f

∂xi

2

+

n

X

i=k+1

∂f

∂xi

2i1

≥ 1

c k x k4

k

X

i=1

xi∂f

∂xi

2

+

n

X

i=k+1

∂f

∂xi

2i

≥ 1

c2 k x k8

By putting C = 1

c 2, α = 8, we have (d)

(d) ⇒ (a) Take g ∈ ε(n), and put ϕ = g − f We have ϕ ∈ m∞

Consider the collection ft(x) = f (x) + tϕ(x), t ∈ [0, 1] By condition d) we have

C, α, δ ≥ 0such that

r

X

i=1

xi∂f

t

∂xi

2

+

n

X

i=r+1

∂ft

∂xi

2

≥ C k x kα, for all k x k< δ, and for all t ∈ [0, 1]

Put

r

X

i=1

xi∂f

t

∂xi

2

+

n

X

i=r+1

∂ft

∂xi

2

= G, and write

−ϕ = G

2(−ϕ)

G2 =

h k

X

i=1

xi

∂ft

∂xi

2

+

n

X

i=k+1

∂ft

∂xi

2i2−ϕ

G2 Put

ξ =h

r

X

i=1

xi∂f

t

∂xi

2

+

n

X

i=r+1

∂ft

∂xi

2i−ϕ

G2 ,

then we have

−ϕ =h

r

X

i=1

xi∂f

t

∂xi

2

+

n

X

i=r+1

∂ft

∂xi

2i ξ,

where ξ is a map which is flat for x and differentiable for t Denote η(x, t) = (η1, , ηn), where

ηi(x, t) = x2i ∂f

t

∂xi
ξi, i = 1, 2, , r and

ηj(x, t) = ∂f

t

∂xi
ξj, j = r + 1, , n

This implies that −ϕ = ∂f t

∂xη(x, t) and ηi is flat for x, and differentiable for t Therefore by the existence of solutions of a differential equation, it follows that for the equation

(

∂h

∂t = η(h(x, t), t), h(x, 0) = x, there exists a unique solution h(x, t), and from η(0, t) = 0 we have h(0, t) = 0 This shows that h(x, t) : (IRn, 0) → (IRn, 0) is deffeomorphism, when small t and

h ∈ Dif fr(n) Because ft(x) = f (x) + tϕ(x), we have ∂f t

∂t = ϕ(x) On the other hand

∂

∂t(f

t◦ h) = ∂f

t

∂x ◦ h∂h

∂t +

∂ft

∂t ◦ h

= ∂f

t

∂x ◦ h.η(h(x, t)) +∂f

t

∂t ◦ h

= −ϕ(h(x, t)) + ∂f

t

∂t ◦ h(x, t)

= 0

This implies that ft◦ his constant for t Thus, we have

f ◦ h(x, 0) = f (x) = f1◦ h(x, 1) = g(h(x, 1))

Therefore, f is ω-determined under the group Diffr(n)  3.2 Remark

(1) If r = 0, then our results coincide results of J Mather, in [1], [5]

(2) If r = 1 and f is a function germ, then we obtain the result of V I Arnold,

in [1]

(3) If r = 1 and f is a map germ, then we have results of H H Vui, in [8] Remark.We have generalized H H Vui's results of the ω-determinancy of map-germs under the group Diffr(n) It is natural to rise that can we generalize the ω-determinancy of map germs under the group K|x , ,x ? where K|x , ,x is

the group, which consisting of all the contact transformations preserving 0 and {x1 = 0}, , {xr = 0}

Acknowledgement The author is indebted to Prof Dr Ha Huy Vui for proposing the problem and for his generous guidance My thanks also to Prof Dr Nguyen Huu Duc for many valuable discussions

References [1] V I Arnold, S M Gusein-Zade, A N Varchenca, Singularities of differentiable maps, Volume I, II, Birkhăauser, Boston-Basel-Stuttgart, 1985.

[2] N T Cuong, N H Duc, N S Minh, H H Vui, Sur les germes de fonctions infiniment determines, C R Aced Sc Paris 285 (1977).

[3] Michael Demazure, Bifurcations and catastrophes, Printed in Germany, Springer-Verlag, Berlin Heidelberg 2000.

[4] Nguyen Tien Dai, Nguyen Huu Duc et Fr´ ed´ eric Pham, Singularites non deg`en`erees des syst`emes Gauss-Manin reticules, Memoire de la societe Mathematique de France, Nouvelle s´ erie No6, Suppl´ ement au Bulletin de la Soci´ et´ e Mat´ ematique de France, (3) 109 (1981) [5] John N Mather, Stability of C ∞ mappings, III Finitely determined map-germs, Publica-tions math´ ematiques de l´I.H ´ E.S., 35 (1968), 127-156.

[6] John N Mather, Stability of C ∞ mappings, II Infinitesimal stability implies stability, Ann Math., 89 (1969), 254-291.

[7] D Siersma, Singularities of functions on boundaries, corners, Quart J Math Oxford (2)

32 (1981), 119 - 127.

[8] H H Vui, Some properties of Taylor of smooth map-germ, Thesis of doctorate, 1980.

tóm tắt

tính ω-xác định của các mầm hàm trên các đa tạp r-reticular Trong bài báo này chúng tôi mở rộng tính ω-xác định của các mầm hàm

f ∈ ε(n)dưới tác động của nhóm Diffr(n)trên đa tạp r-reticular và đưa ra một

điều kiện cần và đủ để một hàm f ∈ ε(n) là ω-xác định dưới tác động của nhóm Dif fr(n) Kết quả chính là Định lý 3.1

(a) Mathematical Section, Department of Education, Tay Nguyen University