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 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, [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 , left-equivalence group , group , contact group , and right-left-equivalence group . In [3, 4], Wilson characterized the infinite determinacy of smooth map-germs with respect to one of the groups , , , and and the infinite determinacy for finitely -determined map-germs with respect to . Besides, for the case of , Brodersen  showed that the results of  hold for map-germs without assuming -finiteness. We can see  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  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,  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 group and relative finite and infinite determinacy with respect to a subgroup of the group 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 and , respectively, both containing the origin. Since this paper is concerned with a local study, without loss of generality, we may assume that
Denote by the space of map-germs , with . Such map-germs are quite common in singularity theory and geometry.
Example 1. Let be given by (right helicoid) It is clear that for and .
Example 2. Let 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 , 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 denote the group of germs at the origin of local diffeomorphisms of , and let where id denotes the identity. Then is a subgroup of .
Let be the local ring , and let denote the maximal ideal of .
Similarly, we can define the corresponding notation for .
Let , where denotes the general linear group, and denotes the identity matrix.
Now, we define two groups
Obviously, and are subgroups of the groups and , respectively. In particular, when , then . The two groups act on the space in the following way.
If , let and ; then and are defined by
Remark 3. Let . If and are -equivalent or -equivalent, then . Set .
In general, if and are -equivalent, then and also are -equivalent. However, this result does not hold for and . For example, let and ; where . Set and , then and . So is -equivalent to . But and are not -equivalent.
Let be the canonical basis of the vector space , and they define a system of generators of -module
For any , then the germ induces a ring homomorphism defined by , for any .
This allows us to consider every -module as an -module via .
Let be the ideal generated by the components , and let denote the image of under , which is not (in general) an ideal of .
For a map-germ , define
The notation and are very nearly the tangent spaces to the orbit of germ under -equivalence and -equivalence, respectively.
Definition 4. Let ; let be a group acting on . We say that the map-germ is --determined if for any germ with , is -equivalent to . If is --determined for some , then it is finitely -determined. If , we say that is --determined.
The rest of this paper is organized as follows. In Section 2, we will give a suffcient condition for --determined map-germs. In Section 3, we study the -determinacy of map-germs. We will give necessary and sufficient conditions for a map-germ to be finitely -determined or --determined.
Throughout the paper, all map-germs will be assumed smooth.
2. The --Determinacy of Map-Germs
In this section, the main result is the following theorem.
Theorem 5. Let 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 . (2).
Then is --determined.
In order to prove this theorem, we need the following results.
Lemma 6 (see ). Let and be a finitely generated -module. Then is finitely generated as a -module if and only if .
Lemma 7. Let . Let be a finitely generated -module. Thenis finitely generated as a -module if and only if
Proof. The proof is essentially the same as that of Corollary in .
Lemma 8. Let and . Then
Proof. For any , then is in . So,
Let be given by .
Since , it follows that
Thus, (11) holds.
Proof of Theorem 5. Suppose that and . Let ; then . For any , define
by and .
Let and denote the map-germs at ; then (or ) induces a ring homomorphism: where (resp., ) denotes the ring of function-germs at which are constant when restricted to (resp., ).
Set and .
Let , where denotes and denotes .
We are trying to show that is -equivalent to . It suffices to show that there exist germs and satisfying the following conditions. (1) and , for any sufficiently close to , and . (2) and . (3), for any sufficiently close to .
By the method initiated by Mather in , it suffices to show that
To see this, it remains to show that (i). (ii).
If (i) and (ii) hold, then
Multiplying (17) by , we get
On the other hand, using condition (1) in Theorem 5, and we get
Obviously, we have
Since , for each , this gives that
Hence, by Nakayama’s lemma we have
From (21), it follows that for any given .
Multiplying (24) by , we get
Thus, (16) follows by substituting (18) in (25).
Now we prove the assertion (i).
By hypothesis, we have for some .
This means that ideal has finite codimension in . Let where and is the projection of in the quotient space,
Now, set ; then is a finitely generated -module. Since ;
Thus, by Lemma 6, is finitely generated -module; that is,
Since and , it follows that
So (i) holds.
Proof of Assertion (ii). Since , by (i) we get
Applying (25) and (i), we have
By (33) and (34), we have
Set , and (35) implies
Now it remains to show that
(a) is a finitely generated -module.
For if this holds, then by Nakayama’s lemma,
To prove (a), by Lemma 7, it suffices to show that
(b) is a finitely generated -module.
Set . By Lemma 8 and (i), we get
Thus, is a -module. Hence, is a -module.
Obviously, is a finitely generated -module; hence is also finitely generated as a -module.
Moreover, since is a finitely generated -module and annihilated by . Thus, is a finitely generated -module. By the argument as equality (29), it follows that is a finitely generated -module. Hence, is a finitely generated -module.
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.
3. The -Determinacy of Map-Germs
In this section, by a similar way as , we will give necessary and sufficient conditions for finitely -determined map-germs and --determined map-germs.
Theorem 9. Suppose that . The following conditions are equivalent. (1) is finitely -determined. (2), for some .
Proof. Let denote the set of -jets at 0 of elements in . If is --determined, then for any with the same -jet as , the -orbit of contains ; that is,
Taking -jets on both sides, we have
Taking tangent spaces at on both sides, we have
By Nakayama’s lemma, this implies
Then (1) implies (2).
Conversely, let be fixed and with . Define by and .
We are trying to show that is -equivalent to . Since and , and is connected, it suffices to show that is -equivalent to , for allsufficiently close to .
Since and , from condition (2), it follows that and
Since is a finitely generated -module, and is the maximal ideal of , by Nakayama’s lemma, we get
Hence also satisfies condition . So where denotes the ring of smooth function-germs in variables at the point .
Let denote the maximal ideal of .
Let . Obviously, is a finitely generated -module.
By the same argument as (46), we have
By (47) and (48), we get
Since , this means that there exist germs in , , such that
Thus, we can find a germ of vector field in of the following form: such that .
That is, we can find a matrix with entries in such that
By integrating the vector field , we get a one-parameter family of local diffeomorphisms in . Thus
Hence, for fixed , is a solution of the differential equation with initial condition .
Since the solution of this differential equation is unique and of the form with as an invertible matrix, and for each . Thus
Thus, and are -equivalent for all sufficiently close to , which completes the proof.
Theorem 10. Suppose that . Then is --determined if and only if
Proof. “Only if”: if is --determined, by the definition, we get
Taking tangent spaces at on both sides, we have
Note that , and . Hence,
“If”: the proof is the same as that of Theorem 9.
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.
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