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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.

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, [

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

Inspired by the aforementioned papers, we will study the relative infinite determinacy for map-germs with respect to a subgroup of the group

Now, we consider the following map-germs.

Let

Denote by

Let

Let

Then

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

Let

Let

Similarly, we can define the corresponding notation for

Let

Now, we define two groups

Obviously,

If

Let

For any

This allows us to consider every

Let

For a map-germ

The notation

Let

The purpose of this paper is to characterize the

The rest of this paper is organized as follows. In Section

Throughout the paper, all map-germs will be assumed smooth.

In this section, the main result is the following theorem.

Let

For some

Then

In order to prove this theorem, we need the following results.

Let

Let

The proof is essentially the same as that of Corollary

Let

For any

Let

Since

Thus, (

Suppose that

Let

Set

Let

We are trying to show that

By the method initiated by Mather in [

To see this, it remains to show that

If (i) and (ii) hold, then

Multiplying (

On the other hand, using condition (1) in Theorem

Obviously, we have

Since

Hence, by Nakayama’s lemma we have

From (

Multiplying (

Thus, (

Now we prove the assertion (i).

By hypothesis, we have

This means that ideal

Now, set

Thus, by Lemma

In particular,

Since

So (i) holds.

Applying (

By (

Set

Now it remains to show that

(a)

For if this holds, then by Nakayama’s lemma,

Hence,

To prove (a), by Lemma

(b)

(c)

Set

Thus,

Obviously,

Moreover, since

The earlier argument shows that

Besides, by (

So (b) and (c) hold. This completes the proof.

In this section, by a similar way as [

Suppose that

Let

Taking

Taking tangent spaces at

By Nakayama’s lemma, this implies

Then (1) implies (2).

Conversely, let

We are trying to show that

Since

Since

Hence

Let

Let

By the same argument as (

By (

Since

Thus, we can find a germ of vector field

That is, we can find a

By integrating the vector field

Hence, for fixed

Since the solution of this differential equation is unique and of the form

Thus,

Suppose that

“Only if”: if

Taking tangent spaces at

Note that

“If”: the proof is the same as that of Theorem

This work was supported by the National Natural Science Foundation of China (Grant no. 11271063) and supported in part by Graduate Innovation Fund of Northeast Normal University of China (no. 12SSXT140). The authors would like to thank the referee for his/her valuable suggestions which improved the first version of the paper.