# Relative Infinite Determinacy for Map-Germs

Authors: Changmei Shi; Donghe Pei
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The infinite determinacy for smooth map-germs with respect to two equivalence relations will be investigated. We treat the space of smooth map-germs with a constraint, and the constraint is that a fixed submanifold in the source space is mapped into another fixed submanifold in the target space. We study the infinite determinacy for such map-germs with respect to a subgroup of right-left equivalence group and finite and infinite determinacy with respect to a subgroup of contact group and give necessary and sufficient conditions for the corresponding determinacy.


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## Body

## 1. Introduction

This work is concerned with the singularity theory of differentiable maps. Singularity theory of differentiable maps is a wide-ranging generalization of the theory of the maxima and minima of functions of one variable, and it is now an essential part of nonlinear analysis. This theory has numerous applications in mathematics and the natural sciences; see, for example, [1, 2].In differentiable analysis, the local behavior of a differentiable map can be determined by the derivatives of the map at a point. Hence we have the theories of finitely and infinitely determined map-germs. We know that every finitely determined map-germ is equivalent to its Taylor polynomial of some degree, and infinite determinacy is a way to express the stability of smooth map-germs under flat perturbations. The analysis of the conditions for a map-germ to be finitely or infinitely determined involves the most important local aspects of singularity theory. Therefore, the study of finite and infinite determinacy of smooth map-germs is a very important subject in singularity theory.Now there are several articles treating the question of infinite determinacy with respect to the most frequently encountered and naturally occurring equivalence groups, for instance, the right-equivalence groupℛ, left-equivalence group ℒ, group 𝒞, contact group 𝒦, and right-left-equivalence group 𝒜. In [3, 4], Wilson characterized the infinite determinacy of smooth map-germs with respect to one of the groups ℛ, ℒ, 𝒞, and 𝒦 and the infinite determinacy for finitely 𝒦-determined map-germs with respect to 𝒜. Besides, for the case of 𝒜, Brodersen [5] showed that the results of [4] hold for map-germs without assuming 𝒦-finiteness. We can see [6] for detailed survey of all of these results. Recently, there appeared the notion of relative infinite determinacy for smooth function-germs with nonisolated singularities. Sun and Wilson [7] treated the smooth function-germs with real isolated line singularities, and this work was later generalized and modified in the case where germs have a nonisolated singular set containing a more general set; for instance, see [8, 9]. In addition, [10] studied the relative versality for map-germs and these map-germs with the constraint that a fixed submanifold in the source space is mapped into another fixed submanifold in the target space.Inspired by the aforementioned papers, we will study the relative infinite determinacy for map-germs with respect to a subgroup of the group𝒜 and relative finite and infinite determinacy with respect to a subgroup of the group 𝒦 by means of some algebraic ideas and tools, and our main results extend the part of the works in [3, 4].Now, we consider the following map-germs.LetN and P be submanifolds without boundary of ℝn and ℝp, respectively, both containing the origin. Since this paper is concerned with a local study, without loss of generality, we may assume that
(1)N={0}×ℝn-n0⊂ℝn,P={0}×ℝp-p0⊂ℝpaaaaaaaaaaaaaaaaaaaaaaaaaaa(n0,p0⩾1).Denote byCnp the space of map-germs f:(ℝn,0)→ℝp, with f(N)⊂P. Such map-germs are quite common in singularity theory and geometry.Example 1.
Letf:(ℝ2,0)→ℝ3 be given by (right helicoid)
(2)f(u,v)=(ucosv,usinv,cv).
It is clear that f(N)⊂P for N={0}×ℝ1⊂ℝ2 and P={0}×ℝ1⊂ℝ3.Example 2.
Letf:(ℝ4,0)→ℝ4 be given by
(3)f(x1,x2,x3,x4)=(x1,x2,x32+x1x4,x42+x2x3).
Thenf(N)⊂P for N=P={0}×ℝ2⊂ℝ4.In this paper we want to characterize the relative infinite determinacy for such map-germs. To formulate the main results we need to introduce some notations and definitions.LetEn denote the ring of smooth function-germs at the origin in ℝn, and let mn denote its unique maximal ideal. For a germ f, let jkf(x) denote the Taylor expansion of f of order k at x. In the case k=∞, j∞f(x) can be identified with the Taylor series of f at x.Letℛ denote the group of germs at the origin of local diffeomorphisms of ℝn, and let
(4)ℛN={ϕ∈ℛ:ϕ|N≡idN},
where id denotes the identity. Then ℛN is a subgroup of ℛ.LetCN be the local ring {f∈En:f|N≡constant}, and let mN denote the maximal ideal of CN.Similarly, we can define the corresponding notation for(ℝp,P).LetℳN={M∣M:(ℝn,0)→GL(p,ℝ)aC∞map-germandM|N=Ip}, where GL(p,ℝ) denotes the general linear group, and Ip denotes the (p×p) identity matrix.Now, we define two groups(5)𝒜N,P=ℛN×ℛP,𝒦N=ℳN×ℛN.Obviously,𝒜N,P and 𝒦N are subgroups of the groups 𝒜 and 𝒦, respectively. In particular, when N=P={0}, then 𝒜N,P=𝒜. The two groups act on the space Cnp in the following way.Iff∈Cnp, let (ϕ,ψ)∈𝒜N,P and (M,h)∈𝒦N; then (ϕ,ψ)·f and (M,h)·f are defined by
(6)(ϕ,ψ)·f=ψ∘f∘ϕ-1,(M,h)·f=M·(f∘h).Remark 3.
(
1
) Let f,g∈Cnp. If f and g are 𝒜N,P-equivalent or 𝒦N-equivalent, then f|N≡g|N. Set ℰ(f,N)={g∈Cnp:f|N≡g|N}.
(
2
) In general, if f and g are 𝒜-equivalent, then f and g also are 𝒦-equivalent. However, this result does not hold for 𝒜N,P and 𝒦N. For example, let f(x1,x2)=(x1,x22) and g(x1,x2)=(x1,x1+x22); where N=P={0}×ℝ1⊂ℝ2. Setϕ=id and ψ(y1,y2)=(y1,y1+y2), then (ϕ,ψ)∈𝒜N,P and (ϕ,ψ)·f=g. So f is 𝒜N,P-equivalent to g. But f and g are not 𝒦N-equivalent.Lete1,…,ep be the canonical basis of the vector space ℝp, and they define a system of generators of CN-module
(7)(CN)×p=CN{e1,…,ep}.For anyf∈Cnp, then the germ f induces a ring homomorphism
(8)f*:CP⟶CN
defined by f*(h)=h∘f, for any h∈CP.This allows us to consider everyCN-module as an CP-module via f*.Letf*mP=〈f1,…,fp0〉 be the ideal generated by the components f1,…,fp0, and let f*(mP) denote the image of mP under f, which is not (in general) an ideal of CN.For a map-germf∈Cnp, define
(9)T𝒜N,Pf=mN〈∂f∂x1,…,∂f∂xn〉+f*(CP){e1,…,ep},T𝒦Nf=mN〈∂f∂x1,…,∂f∂xn〉+f*mp·(mN)×p.The notationT𝒜N,Pf and T𝒦Nf are very nearly the tangent spaces to the orbit of germ f under 𝒜N,P-equivalence and 𝒦N-equivalence, respectively.Definition 4.
Letf∈Cnp; let G be a group acting on Cnp. We say that the map-germ f is k-G-determined if for any germ g∈ℰ(f,N) with jkf(0)=jkg(0), g is G-equivalent to f. If f is k-G-determined for some k<∞, then it is finitely G-determined. If k=∞, we say that f is ∞-G-determined.The purpose of this paper is to characterize the∞-𝒜N,P-determinacy and ∞-𝒦N-determinacy for map-germs. The main results in this paper are stated in Theorems 5 and 10.The rest of this paper is organized as follows. In Section2, we will give a suffcient condition for ∞-𝒜N,P-determined map-germs. In Section 3, we study the 𝒦N-determinacy of map-germs. We will give necessary and sufficient conditions for a map-germ to be finitely 𝒦N-determined or ∞-𝒦N-determined.Throughout the paper, all map-germs will be assumed smooth.


## 2. The∞-𝒜N,P-Determinacy of Map-Germs

In this section, the main result is the following theorem.Theorem 5.
Letf∈Cnp be a map-germ. Suppose that f satisfies the following conditions. (1)
For somer<∞, mnrmN⊂mNJ(f)+f*mP·mN, where J(f) denotes the ideal in En generated by the determinants of (p×p) minors of the Jacobian matrix df of f.(2)
(mn∞mN)×p⊂T𝒜N,Pf.
Thenf is ∞-𝒜N,P-determined.In order to prove this theorem, we need the following results.Lemma 6 (see [10]).
Letf∈Cnp and M be a finitely generated CN-module. Then M is finitely generated as a f*(CP)-module if and only if dimℝ(M/f*mP·M)<∞.Lemma 7.
Letf,g∈Cnp. Let M be a finitely generated (f,g)*(CP×P)-module. ThenMis finitely generated as a g*(CP)-module if and only if
(10)dimℝMg*mP·M<∞.Proof.
The proof is essentially the same as that of Corollary1.17 in [11].Lemma 8.
Letf,g∈Cnp and g-f∈(mn∞mN)×p. Then
(11)(f,g)*(CP×P)⊂f*(CP)+mn∞mN.Proof.
For anyh∈CP×P, then (f,g)*(h)-(f,f)*(h) is in mn∞mN. So,
(12)(f,g)*(CP×P)⊂(f,f)*(CP×P)+mn∞mN.
Letϕ:(ℝp,P)→(ℝp×ℝp,P×P) be given by ϕ(y)=(y,y).
Since(f,f)*(h)=f*(ϕ*(h)), it follows that
(13)(f,f)*(CP×P)⊂f*(CP).
Thus, (11) holds.Proof of Theorem5.
Suppose thatf~∈ℰ(f,N) and j∞f~(0)=j∞f(0). Let u=f~-f; then u∈(mn∞mN)×p. For any t0∈[0,1], define
(14)g:(ℝn×ℝ,(0,t0))⟶ℝp
by g(x,t)=f(x)+tu(x) and gt(x)=g(x,t).
LetF(x,t)=(f(x),t) and G(x,t)=(g(x,t),t) denote the map-germs at (0,t0); then F (or G) induces a ring homomorphism:
(15)F*:CP×{t0}⟶CN×{t0},h⟼h∘F=F*(h),
where CN×{t0} (resp., CP×{t0}) denotes the ring of function-germs at (0,t0) which are constant when restricted to N×{t0} (resp., P×{t0}).
SetmNkCN×{t0}=mTk and mPkCP×{t0}=mT′k.
LetT-𝒜N,PF=tF+ωF, where tF denotes mT〈∂f/∂x1,…,∂f/∂xn〉 and ωF denotes F*(CP×{t0}){e1,…,ep}.
We are trying to show thatf~ is 𝒜N,P-equivalent to f. It suffices to show that there exist germs ϕ:(ℝn×ℝ,(0,t0))→(ℝn,0) and ψ:(ℝp×ℝ,(0,t0))→(ℝp,0) satisfying the following conditions. (1)
ϕ(x,t)=x and ψ(y,t)=y, for any t sufficiently close to t0, x∈N and y∈P.(2)
ϕt0=idℝn and ψt0=idℝp.(3)
gt0=ψt∘gt∘ϕt-1, for any t sufficiently close to t0.
By the method initiated by Mather in [12], it suffices to show that
(16)(mn∞mT)×p⊂mT〈∂g∂x1,…,∂g∂xn〉+G*(mT′){e1,…,ep}.
To see this, it remains to show that(i)
(mn∞mT)×p⊂T-𝒜N,PF.(ii)
T-𝒜N,PF=T-𝒜N,PG.
If (i) and (ii) hold, then(17)(mn∞mT)×p⊂T-𝒜N,PG.
Multiplying (17) by G*mT′, we get
(18)G*mT′·(mn∞mT)×p⊂mT〈∂g∂x1,…,∂g∂xn〉+G*(mT′){e1,…,ep}.
On the other hand, using condition (1) in Theorem5, and
(19)J(f)·(mN)×p⊂mN〈∂f∂x1,…,∂f∂xn〉,
we get
(20)(mnrmN)×p⊂mN〈∂f∂x1,…,∂f∂xn〉+f*mP·(mN)×p.
Obviously, we have(21)(mn∞mN)×p⊂mN〈∂f∂x1,…,∂f∂xn〉+f*mP·(mN)×p.
Sincegt-f∈(mn∞mN)×p, for each t∈ℝ, this gives that
(22)mN〈∂g∂x1,…,∂g∂xn〉+gt*mP·(mN)×p⊂mN〈∂f∂x1,…,∂f∂xn〉+f*mP·(mN)×p,mN〈∂f∂x1,…,∂f∂xn〉+f*mP·(mN)×p⊂mN〈∂g∂x1,…,∂g∂xn〉+gt*mP·(mN)×p+mn(mn∞mN)×p.
Hence, by Nakayama’s lemma we have(23)mN〈∂f∂x1,…,∂f∂xn〉+f*mP·(mN)×p=mN〈∂g∂x1,…,∂g∂xn〉+gt*mP·(mN)×p.
From (21), it follows that
(24)(mn∞mN)×p⊂mN〈∂g∂x1,…,∂g∂xn〉+gt*mP·(mN)×p,
for any given t∈ℝ.
Multiplying (24) by mn∞CN×{t0}, we get
(25)(mn∞mT)×p⊂mT〈∂g∂x1,…,∂g∂xn〉+G*mT′·(mn∞mT)×p.
Thus, (16) follows by substituting (18) in (25).
Now we prove the assertion (i).
By hypothesis, we have(26)mnrmN⊂mNJ(f)+f*mP·CN,
for some r<∞.
This means that idealmNJ(f)+f*mP·CN has finite codimension in CN. Let
(27)CN(mNJ(f)+f*mP·CN)=ℝ{a-1,…,a-s},
where ai∈CN and a-i is the projection of ai in the quotient space, i=1,…,s.
Now, setM=CN×{t0}/mTJ(f); then M is a finitely generated CN×{t0}-module. Since F*mP×{t0}·CN×{t0}=f*mP·CN×{t0}+(t-t0)CN×{t0};
(28)MF*mP×{t0}·M≅CNmNJ(f)+f*mP·CN.
Thus, by Lemma6, M is finitely generated F*(CP×{t0})-module; that is,
(29)M=F*(CP×{t0}){a1,…,as}.
In particular,(30)CN×{t0}=mTJ(f)+F*(CP×{t0})CN.
Since(31)T-𝒜N,PF⊃(mN〈∂f∂x1,…,∂f∂xn〉+F*(CP×{t0}){e1,…,ep}∂f∂x1)·F*(CP×{t0})⊃T𝒜N,Pf·F*(CP×{t0})⊃(mn∞mN)×p·F*(CP×{t0}),
and mNJ(f)·(CN×{t0})×p⊂mT〈∂f/∂x1,…,∂f/∂xn〉, it follows that
(32)T-𝒜N,PF⊃(mn∞mN)×p·[mTJ(f)+F*(CP×{t0})CN]=(mn∞mT)×p.
So (i) holds.
Proof of Assertion (ii). Since g-f∈(mn∞mT)×p, by (i) we get
(33)T-𝒜N,PG⊆T-𝒜N,PF=T-𝒜N,PG+(mn∞mT)×p.
Applying (25) and (i), we have
(34)(mn∞mT)×p⊂mT〈∂g∂x1,…,∂g∂xn〉+G*mP×{t0}·(mn∞mT)×p⊂mT〈∂g∂x1,…,∂g∂xn〉+G*mP×{t0}·T-𝒜N,PF.
By (33) and (34), we have
(35)T-𝒜N,PG⊆T-𝒜N,PF=T-𝒜N,PG+G*mP×{t0}·T-𝒜N,PF.
SetE=T-𝒜N,PF/tG, and (35) implies
(36)E=T-𝒜N,PGtG+G*mP×{t0}·E.
Now it remains to show that
(a)E is a finitely generated G*(CP×{t0})-module.
For if this holds, then by Nakayama’s lemma,(37)E=T-𝒜N,PGtG.
Hence,T-𝒜N,PF=T-𝒜N,PG.
To prove (a), by Lemma7, it suffices to show that
(b)E is a finitely generated (F,G)*(CP×{t0}×P×{t0})-module.
(c)dimℝ(E/G*mP×{t0}·E)<∞.
SetR=(F,G)*(CP×{t0}×P×{t0}). By Lemma 8 and (i), we get
(38)R·T-𝒜N,PF⊂[F*(CP×{t0})+mn∞mT]·T-𝒜N,PF⊂T-𝒜N,PF.
Thus,T-𝒜N,PF is a R-module. Hence, E is a R-module.
Obviously,ωF is a finitely generated F*(CP×{t0})-module; hence ωF is also finitely generated as a R-module.
Moreover, sincetF/tG∩tF is a finitely generated CN×{t0}-module and annihilated by mTJ(gt). Thus, tF/tG∩tF is a finitely generated CN×{t0}/mTJ(gt)-module. By the argument as equality (29), it follows that CN×{t0}/mTJ(gt) is a finitely generated G*(CP×{t0})-module. Hence, tF/tG∩tF is a finitely generated R-module.
The earlier argument shows thatE is finitely generated as a R-module.
Besides, by (35), we have
(39)dimℝEG*mP×{t0}·E=dimℝT-𝒜N,PG+G*mP×{t0}·T-𝒜N,PFtG+G*mP×{t0}(T-𝒜N,PG+T-𝒜N,PF)=dimℝωG+(tG+G*mP×{t0}·T-𝒜N,PF)G*mP×{t0}·ωG+(tG+G*mP×{t0}·T-𝒜N,PF)≤dimℝωGG*mP×{t0}·ωG<∞.
So (b) and (c) hold. This completes the proof.


## 3. The𝒦N-Determinacy of Map-Germs

In this section, by a similar way as [13], we will give necessary and sufficient conditions for finitely 𝒦N-determined map-germs and ∞-𝒦N-determined map-germs.Theorem 9.
Suppose thatf∈Cnp. The following conditions are equivalent. (1)
f is finitely 𝒦N-determined.(2)
(mnrmN)×p⊂T𝒦Nf, for some r∈ℕ.Proof.
LetJr+1(n,p) denote the set of (r+1)-jets at 0 of elements in ℰ(f,N). If f is r-𝒦N-determined, then for any g∈ℰ(f,N) with the same r-jet as f, the 𝒦N-orbit of f contains g; that is,
(40){g∈ℰ(f,N):jrg(0)=jrf(0)}⊂{g∈ℰ(f,N):gandfare𝒦N-equivalent}.
Taking(r+1)-jets on both sides, we have
(41){jr+1g(0)∈Jr+1(n,p):jrg(0)=jrf(0)}⊂{jr+1g(0)∈Jr+1(n,p):gandfare𝒦N-equivalentjr+1}.
Taking tangent spaces atjr+1f(0) on both sides, we have
(42)(mnrmN)×p⊂T𝒦Nf+(mnr+1mN)×p.
By Nakayama’s lemma, this implies(43)(mnrmN)×p⊂T𝒦Nf.
Then (1) implies (2).
Conversely, lett0∈[0,1] be fixed and g∈ℰ(f,N) with jr+1f(0)=jr+1g(0). Define
(44)F:(ℝn×ℝ,(0,t0))⟶ℝp
by F(x,t)=f(x)+t(g(x)-f(x)) and Ft(x)=F(x,t).
We are trying to show thatg is 𝒦N-equivalent to f. Since F0=f and F1=g, and [0,1] is connected, it suffices to show that Ft is 𝒦N-equivalent to Ft0, for alltsufficiently close to t0.
SinceFt0(x)-f(x)=t0(g(x)-f(x)) and g-f∈(mnr+1mN)×p, from condition (2), it follows that
(45)T𝒦NFt0⊂T𝒦Nf,
and T𝒦Nf⊂T𝒦NFt0+mn·(T𝒦Nf).
SinceT𝒦Nf is a finitely generated En-module, and mn is the maximal ideal of En, by Nakayama’s lemma, we get
(46)T𝒦NFt0=T𝒦Nf.
HenceT𝒦NFt0 also satisfies condition (2). So
(47)(mnrmNEn+1*)×p⊂T𝒦NFt0·En+1*,
where En+1* denotes the ring of smooth function-germs in variables (x,t) at the point (0,t0).
Letmn+1* denote the maximal ideal of En+1*.
LetT-𝒦NF=mNEn+1*〈∂F/∂x1,…,∂F/∂xn〉+F*mp·(mNEn+1*)×p. Obviously, T-𝒦NF is a finitely generated En+1*-module.
By the same argument as (46), we have
(48)T-𝒦NF=T𝒦NFt0·En+1*.
By (47) and (48), we get
(49)(mnrmNEn+1*)×p⊂T-𝒦NF.
Since∂F/∂t=g-f∈(mnr+1mN)×p, this means that there exist germs Xi in mNEn+1*, i=1,…,n, such that
(50)(g-f)+∑i=1n‍Xi(x,t)∂F∂xi∈F*mp·(mNEn+1*)×p.
Thus, we can find a germ of vector fieldX in ℝn×ℝ of the following form:
(51)∂∂t+∑i=1n‍Xi(x,t)∂∂xi,
such that DF(X)∈F*mp·(mNEn+1*)×p.
That is, we can find a(p×p) matrix A(x,t) with entries in mNEn+1* such that
(52)DF(X)=A(x,t)·F(x,t).
By integrating the vector fieldX, we get a one-parameter family of local diffeomorphisms ϕt in ℛN. Thus
(53)ddtF(ϕt(x),t)=∂∂tF(ϕt(x),t)+∑i=1n‍Xi(ϕt(x),t)∂F∂xi(ϕt(x),t)=A(ϕt(x),t)·F(ϕt(x),t).
Hence, for fixedx∈ℝn, F(ϕt(x),t) is a solution of the differential equation y˙=A(ϕt(x),t)y with initial condition y(x,t0)=Ft0(x).
Since the solution of this differential equation is unique and of the form(54)y(x,t)=B(x,t)·y(x,t0),
with B(x,t) as an invertible matrix, B(x,t0)=Ip and B(x,t)=Ip for each x∈N. Thus
(55)Ft(x)=B(ϕt-1(x),t)·(Ft0∘ϕt-1(x)).
Thus,Ft and Ft0 are 𝒦N-equivalent for all t sufficiently close to t0, which completes the proof.Theorem 10.
Suppose thatf∈Cnp. Then f is ∞-𝒦N-determined if and only if
(56)(mn∞mN)×p⊂T𝒦Nf.Proof.
“Only if”: iff is ∞-𝒦N-determined, by the definition, we get
(57)f+(mn∞mN)×p⊂𝒦N·f.
Taking tangent spaces atf on both sides, we have
(58)Tf(f+(mn∞mN)×p)⊂Tf(𝒦N·f).
Note that(mn∞mN)×p⊂Tf(f+(mn∞mN)×p), and Tf(𝒦N·f)⊂T𝒦Nf. Hence,
(59)(mn∞mN)×p⊂T𝒦Nf.
“If”: the proof is the same as that of Theorem9.



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*Source: 549845-2013-10-22.xml*