Relative Infinite Determinacy for Map-Germs

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 suchmap-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.


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 group R, left-equivalence group L, group C, contact group K, and right-leftequivalence 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 mapgerms 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 group A 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.
Let  and  be submanifolds without boundary of R  and R  , respectively, both containing the origin.Since this paper Journal of Function Spaces and Applications is concerned with a local study, without loss of generality, we may assume that Denote by    the space of map-germs  : (R  , 0) → R  , with () ⊂ .Such map-germs are quite common in singularity theory and geometry.
Example 1.Let  : (R 2 , 0) → R 3 be given by (right helicoid) ( It is clear that Example 2. Let  : (R 4 , 0) → R 4 be given by Then () ⊂  for 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.
Let   denote the ring of smooth function-germs at the origin in R  , and let   denote its unique maximal ideal.For a germ , let   () denote the Taylor expansion of  of order  at .In the case  = ∞,  ∞ () can be identified with the Taylor series of  at .
Let R denote the group of germs at the origin of local diffeomorphisms of R  , and let where id denotes the identity.Then R  is a subgroup of R.
Let   be the local ring { ∈   : |  ≡ constant}, and let   denote the maximal ideal of   .
Similarly, we can define the corresponding notation for (R  , ).
Let M  = { |  : (R  , 0) → GL(, R) a  ∞ map-germ and |  =   }, where GL(, R) denotes the general linear group, and   denotes the ( × ) identity matrix.Now, we define two groups Obviously, A , and K  are subgroups of the groups A and K, respectively.In particular, when  =  = {0}, then A , = A. The two groups act on the space    in the following way.
If  ∈    , let (, ) ∈ A , and (, ℎ) ∈ K  ; then (, ) ⋅  and (, ℎ) ⋅  are defined by (2) In general, if  and  are A-equivalent, then  and  also are K-equivalent.However, this result does not hold for A , and K  .For example, let ( 1 ,  2 ) = ( 1 ,  2  2 ) and Let  1 , . . .,   be the canonical basis of the vector space R  , and they define a system of generators of   -module For any  ∈    , then the germ  induces a ring homomorphism defined by  * (ℎ) = ℎ ∘ , for any ℎ ∈   .This allows us to consider every   -module as an  module via  * .
For a map-germ  ∈ The notation A ,  and K   are very nearly the tangent spaces to the orbit of germ  under A , -equivalence and K  -equivalence, respectively.Definition 4. Let  ∈    ; let  be a group acting on    .We say that the map-germ  is --determined if for any germ The purpose of this paper is to characterize the ∞-A ,determinacy and ∞-K  -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 Section 2, we will give a suffcient condition for ∞-A , -determined map-germs.In Section 3, we study the K  -determinacy of map-germs.We will give necessary and sufficient conditions for a map-germ to be finitely K  -determined or ∞-K determined.
Throughout the paper, all map-germs will be assumed smooth.

The ∞-A 𝑁,𝑃 -Determinacy of Map-Germs
In this section, the main result is the following theorem.

Theorem 5. Let 𝑓 ∈ 𝐶 𝑝
be a map-germ.Suppose that  satisfies the following conditions.
(1) For some  < ∞,      ⊂   () +  *   ⋅   , where () denotes the ideal in   generated by the determinants of ( × ) minors of the Jacobian matrix  of . ( In order to prove this theorem, we need the following results. Lemma 6 (see [10]).Let  ∈ Proof.The proof is essentially the same as that of Corollary 1.17 in [11].
By the method initiated by Mather in [12], it suffices to show that To see this, it remains to show that (ii) A ,  = A , .
The earlier argument shows that  is finitely generated as a -module.
Besides, by (35), we have So (b) and (c) hold.This completes the proof.

The K 𝑁 -Determinacy of Map-Germs
In this section, by a similar way as [13], we will give necessary and sufficient conditions for finitely K  -determined mapgerms and ∞-K  -determined map-germs.
Theorem 9. Suppose that  ∈    .The following conditions are equivalent.
We are trying to show that  is K  -equivalent to .Since  0 =  and  1 = , and [0, 1] is connected, it suffices to show that   is K  -equivalent to   0 , for all  sufficiently close to  0 .