1 Introduction

Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

\ArticleName

Nonlinear Stability of Relative Equilibria
and Isomorphic Vector Fields

\Author

Department of Mathematics, University of Illinois at Urbana-Champaign,
1409 W. Green Street, Urbana, IL 61801 USA \Emailklajbor2@illinois.edu \URLaddresshttps://faculty.math.illinois.edu/~klajbor2/

\ArticleDates

Received October 31, 2017, in final form March 09, 2018; Published online March 14, 2018

\Abstract

We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542–587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos’s criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.

\Keywords

equivariant dynamics; relative equilibria; orbital stability; Hamiltonian systems

\Classification

37J25; 57R25; 37J15; 53D20

## 1 Introduction

Relative equilibria of equivariant vector fields and their stability have garnered much interest in the dynamics literature, partly due to their myriad applications in the sciences (see, for example, [6]). In this paper we present an approach to determining the stability of relative equilibria via the notion of isomorphic vector fields introduced by Hepworth [9]. In particular, we argue that it can be useful to replace a given equivariant vector field with an isomorphic one for which it is easier to determine stability.

Recall that a relative equilibrium of an equivariant vector field is a point for which the vector field is tangent to the group orbit at that point. It can be difficult to determine the stability of relative equilibria. Even determining linear stability poses a challenge. For an equilibrium, the Lyapunov stability criterion can guarantee linear stability if all the eigenvalues in the spectrum of the linearization of the vector field have negative real part (see, for example, [1, Theorem 4.3.4]). In contrast, since the vector field is not necessarily zero at a relative equilibrium, the usual notion of a linearization does not make sense. Thus, we don’t immediately have an analogue of the Lyapunov stability criterion.

A construction due to Krupa gives a way to linearize an equivariant vector field near a relative equilibrium and test for linear stability [11]. Krupa’s construction involves choosing a slice for the action through the relative equilibrium and projecting the vector field onto the slice. The projected vector field has an equilibrium at the original vector field’s relative equilibrium, so we can linearize the projected vector field. This construction depends on a choice of slice and projection, but it turns out the real parts of the spectrum of the linearization are independent of these choices [5, Lemma 8.5.2]. The Lyapunov stability criterion can then be used to test for linear stability of the equilibrium of the projected vector field. It can be shown that if this is linearly stable it implies the linear stability of the relative equilibrium of the original vector field [2, Theorem 7.4.2]. Furthermore, since the real parts of the eigenvalues of the spectrum are independent of the choices, we can choose any slice and projection to determine linear stability; ideally ones where the spectrum is easier to compute.

Not all stable equilibria are linearly stable, and the same is true of relative equilibria. To use Krupa’s construction for nonlinear stability, as well as for other applications, we need to make sense of the choices involved. Hepworth’s notion of isomorphism of vector fields is useful for this. Hepworth introduced isomorphic vector fields to define vector fields on differentiable stacks, a categorical generalization of differentiable manifolds [9]. Since differentiable stacks are, in some sense, represented by Lie groupoids, it is not surprising that vector fields on a stack form a groupoid. This gives rise to a notion of isomorphism between equivariant vector fields. Lerman used Hepworth’s notion of isomorphism of vector fields to revisit Krupa’s construction [12]. In particular, he showed that the choices of slice and projection lead to isomorphic vector fields.

In this paper we show how considering vector fields up to isomorphism, in the sense of Hepworth, facilitates testing for nonlinear stability. In Theorem 3.11, which we call here the slice stability criterion, we show that one can determine nonlinear stability of a relative equilibrium by testing for nonlinear stability of the corresponding equilibrium of the projected vector field. This reduces the problem to the well-studied case of equilibria on a vector space with a representation of a compact Lie group. In fact, one can test any vector field that is isomorphic to the projected vector field. Hence, one additionally obtains the freedom to choose a convenient slice, projection, and isomorphism class representative to determine stability.

Hamiltonian relative equilibria are an important case where we may have nonlinear stability but not linear stability. The integral curves of a Hamiltonian vector field do not exhibit energy dissipation, so we don’t expect the relative equilibria to be linearly stable. Lerman and Singer [13] and Ortega and Ratiu [18], building on work of Patrick [21, 22], showed that the definiteness of the Hessian of an augmented Hamiltonian function implies stability of the Hamiltonian relative equilibrium. Montaldi and Rodríguez-Olmos extended this criterion, allowing for a wide choice of augmented Hamiltonians to check for stability [16, Theorem 3.6] (see also [17, Theorem 2]). They prove this extension by building on the bundle equations in [23, 24, 25]. We use Theorem 3.11 to provide an alternative proof of their result. Our proof is based on the fact that the augmented Hamiltonian vector fields are isomorphic to the original Hamiltonian vector field and that a choice of augmented Hamiltonian is equivalent to a choice of an isomorphism class and a representative.

### 1.1 Organization of the paper

In Section 2, we present Hepworth’s groupoid of equivariant vector fields in the context of Lie group actions, as well as the corresponding notion of isomorphism of equivariant vector fields. We also present an equivalent formulation of the results in [12], and provide some general background and results.

In Section 3, we prove a test for nonlinear stability of relative equilibria, Theorem 3.11, which we call here the slice stability criterion. This is our main theorem on the nonlinear stability of relative equilibria. We also show how isomorphisms of equivariant vector fields and one of the functors involved in the slice stability criterion preserve the stability of relative equilibria.

In Section 4, we apply the slice stability criterion to obtain a proof of the result of Montaldi and Rodríguez-Olmos (Theorem 4.8). We use the Marle–Guillemin–Sternberg normal form [8, 14] in this proof. In Section 5, we reduce the general case to the normal form computation.

### 1.2 Notation and conventions

Throughout the paper we will assume all manifolds are Hausdorff. We will denote Lie groups with uppercase Latin letters, their Lie algebras with the corresponding lowercase fraktur letter, and the duals of these Lie algebras by adding a star superscript. The adjoint representation of a Lie group on its Lie algebra will be denoted by , while its coadjoint representation on the dual of the Lie algebra will be denoted by . Given an action of a Lie group on a manifold, the stabilizer subgroup of a point will be denoted by the same letter as the group but with the point as a subscript (e.g., ). The Lie algebra of the stabilizer will also carry the point as a subscript (e.g., ).

The vector space of smooth vector fields on a manifold will be denoted by . Given a diffeomorphism between two manifolds, we will denote the corresponding pushforward of vector fields along by and the pullback of vector fields along  by . We will refer to both embedded and regular submanifolds. Recall an embedded submanifold of a manifold is a pair consisting of a manifold and a smooth embedding , whereas a regular submanifold of a manifold  consists of a subset  of , with smooth charts adapted from the charts of , for which the inclusion is a smooth embedding.

Given a smooth fiber bundle , the corresponding vertical bundle is the bundle over the manifold with total space . The projection is the restriction of the tangent bundle projection , and hence the vertical bundle is a subbundle of the tangent bundle. We will also make use of associated bundles. Given a Lie group , a manifold  with a free and proper right action of , and a manifold  with a proper left action of , the associated bundle is the bundle over the smooth orbit space with total space . Here, the group acts on the space by in a free and proper fashion from the left. We will denote the elements of by . The bundle projection is defined by , where is the -orbit of . If the manifold is a product of the form , we will denote the elements of by instead of .

## 2 Relative equilibria and isomorphic vector fields

In this section we define the groupoid of equivariant vector fields on a manifold with a group action, and the corresponding notion of isomorphism of equivariant vector fields. We then describe Krupa’s construction in this language, and Lerman’s results about the groupoids of equivariant vector fields present in this construction. Along the way, we discuss how relative equilibria are preserved by isomorphisms of equivariant vector fields, equivariant extension of vector fields, pushforward and pullbacks of vector fields (when these are defined), and certain functors between groupoids of equivariant vector fields.

We work in the following setting:

###### Definition 2.1 (G-manifold).

A manifold with an action of a Lie group is called a -manifold. If the action of is a proper action then we say is a proper -manifold.

By an equivariant vector field we mean:

###### Definition 2.2 (equivariant vector field).

A vector field on a manifold is equivariant with respect to the action of a Lie group if for all we have , where is the diffeomorphism . If we need to specify the group we say is -equivariant.

We next recall the definition of a relative equilibrium:

###### Definition 2.3 (relative equilibrium).

Given an equivariant vector field on a -manifold , a point is a relative equilibrium of if the vector is tangent to the group orbit . If we need to specify the group we say is a -relative equilibrium.

###### Definition 2.4 (velocities).

Let be a proper -manifold, let be an equivariant vector field on , and let be a point in . A velocity for the point is a vector such that , where

 ξM: M→TM,ξM(m):=ddt∣∣0exp(tξ)⋅m

is the fundamental vector field generated by the vector .

###### Remark 2.5.

Velocities exist for relative equilibria since

 Tm(G⋅m)={νM(m)|ν∈g}.

In fact, the existence of velocities at a point characterize that point as a relative equilibrium. Furthermore, since

 gm={η∈g|ηM(m)=0},

velocities are unique modulo the Lie algebra of the stabilizer.

The following maps are needed to define morphisms of vector fields:

###### Definition 2.6 (infinitesimal gauge transformations).

Infinitesimal gauge transformations are the elements of the vector space

###### Remark 2.7.

If the action of on is free and proper, then the orbit space is a manifold and the orbit space map is a principal -bundle. In this case, the space of infinitesimal gauge transformations is isomorphic to the space of smooth sections of the bundle .

The space of infinitesimal gauge transformations acts, as a group under pointwise addition, on the space of equivariant vector fields by

 C∞(M,g)G×Γ(TM)G→Γ(TM)G,(ψ,X)↦X+ψM,

where denotes the vector field on defined by

 ψM: M→TM,ψM(m):=ddt∣∣0exp(tψ(m))⋅m.
###### Lemma 2.8.

Let be a -manifold and let be an infinitesimal gauge transformation on . The induced vector field is an equivariant vector field with respect to the action of .

###### Proof.

This is a consequence of the naturality of the exponential. Let and , then

 ψM(g⋅m) =ddt∣∣0exp(tψ(g⋅m))⋅g⋅m =ddt∣∣0exp(tAd(g)ψ(m))⋅g⋅m by the equivariance of ψ =ddt∣∣0gexp(tψ(m))g−1⋅g⋅m by the naturality of exp =(dgM)mψM(m).

Hence, is an equivariant vector field. ∎

We can now define the groupoid of equivariant vector fields:

###### Definition 2.9 (groupoid of equivariant vector fields).

Let be a -manifold. The groupoid of equivariant vector fields is the action groupoid corresponding to the action of the infinitesimal gauge transformations on the -equivariant vector fields . The groupoid of -equivariant vector fields has

 objects:equivariant vector fields X∈Γ(TM)G, morphisms:pairs (ψ,X)∈C∞(M,g)G×Γ(TM)G.

The source function is given by

 s: C∞(M,g)G×Γ(TM)G→Γ(TM)G,(ψ,X)↦X,

and the target function is given by

 t: C∞(M,g)G×Γ(TM)G→Γ(TM)G,(ψ,X)↦X+ψM.

The composition of a composable pair of morphisms and is given by

 (φ,X+ψM)∘(ψ,X)=(φ+ψ,X).

The unit function is given by

 u: Γ(TM)G→C∞(M,g)G×Γ(TM)G,X↦(0,X),

and the inversion function is given by

 −: C∞(M,g)G×Γ(TM)G→C∞(M,g)G×Γ(TM)G,(ψ,X)↦(−ψ,X).
###### Remark 2.10.

Hepworth [9] defined vector fields on differentiable stacks and showed they form a category. In the case of a quotient stack for the action of a compact group on a manifold , Hepworth showed that the category of vector fields on the stack is equivalent to the category given in Definition 2.9 [9, Proposition 5.1].

In the following definition we highlight what it means for two vector fields to be isomorphic in the groupoid of equivariant vector fields:

###### Definition 2.11 (isomorphic vector fields).

Two equivariant vector fields and on a -manifold are -isomorphic if there exists an infinitesimal gauge transformation  in the space such that

 Y=X+ψM.

As noted in [12, Corollary 2.8], isomorphisms of equivariant vector fields preserve relative equilibria in the following sense:

###### Lemma 2.12.

Let and be two isomorphic equivariant vector fields on a -manifold . If a point is a relative equilibrium of then it is a relative equilibrium of .

###### Proof.

Since and are isomorphic, there exists a map such that . Note that the vector is tangent to the group orbit since the point is a relative equlibrium of . The vector is also tangent to the group orbit since the vector is defined to be the derivative of a curve on the group orbit of the point . Thus, we have

 Y(m)=X(m)+ψM(m)∈Tm(G⋅m),

meaning the point is a relative equilibrium of the vector field . ∎

We will use the following vector fields in our application of Theorem 3.11 to Hamiltonian relative equilibria in Section 4:

###### Definition 2.13 (augmented vector fields).

Let be a proper -manifold and an equivariant vector field on . Given a vector , the corresponding vector field augmented by is the vector field

 Xξ: M→TM,Xξ:=X−ξM.
###### Remark 2.14.

Given a -equivariant vector field on a proper -manifold , the corresponding augmented vector field is not -equivariant. However, it is equivariant with respect to the Lie subgroup

Also note, that if is a velocity for a -relative equilibrium of the vector field , then the augmented vector field has an equilibrium at the point .

###### Lemma 2.15.

Let be a proper -manifold, let be an equivariant vector field on , and let be a given vector in the Lie algebra of . The vector field is -isomorphic to its augmented vector field .

###### Proof.

Let be the Lie algebra of the Lie subgroup . The constant map

 ξ: M→gξ,m↦ξ

is a smooth -equivariant map, and hence gives a morphism of the groupoid of -equivariant vector fields . Note by definition, so the result follows. ∎

Recall we can assemble the maximal integral curves of a smooth vector field on a Hausdorff manifold into a maximal flow:

###### Definition 2.16 (Flow).

Let be a Hausdorff manifold and let be a smooth vector field on . For every point , let be the maximal integral curve of such that . Let be the open subset of defined by

 A:=⋃m∈MIm×{m}.

The maximal flow, or just flow, of the vector field is the smooth map

 ϕ: A→M,ϕ(t,m):=γm(t).

The set is called the flow domain of .

###### Remark 2.17.

It is important to recall that we are assuming all manifolds are Hausdorff, this is required for some of the definitions and results in this paper. From now on, we won’t explicitly mention this hypothesis.

The following result, due to Lerman, relates the flows of isomorphic vector fields:

###### Theorem 2.18 (Lerman [12, Theorem 1.6]).

Let be a proper -manifold and let and be two isomorphic equivariant vector fields on . Then there exists a family of smooth maps depending smoothly on so that the maximal flows and , of and respectively, satisfy

 ϕX(t,m)=Ft(m)⋅ϕY(t,m)

for all in the domain of the flow .

We recall the following notion of continuous flows on topological spaces:

###### Definition 2.19.

Let be a topological space, and let be an open subset of containing the set . An abstract flow on is a continuous map satisfying:

1. for all ;

2. whenever both sides make sense.

For a given point , the curve of the abstract flow starting at is the curve defined by , where consists of all times for which .

We recall the following standard result about -equivariant vector fields:

###### Lemma 2.20.

Let be a proper -manifold and let be an equivariant vector field on . The flow of the vector field induces an abstract flow on the orbit space such that the following diagram commutes:

 \xy(−14,10)∗+A="1";(12,10)∗+M="2";(−14,−6)∗B="3";(12,−6)∗+M/G,="4";\ar@−>ϕ(−12,10);(8,10);\ar@−>id×π(−14,7.5);(−14,−3);\ar@−>π"2";"4";\ar@−>Φ"3";"4";\endxy (2.1)

where is the orbit map.

###### Proof.

First, define the set and the map

 Φ: B→M/G,Φ(t,π(m)):=π(ϕ(t,m)).

We want to show that the map is our desired abstract flow. Thus, we need to show that the set is open, that contains , that the map is well-defined and continuous, that  makes the diagram (2.1) commute, and that satisfies properties (1) and (2) in Definition 2.19.

Observe that the action of the Lie group on the manifold gives an action on the product by

 g⋅(t,m):=(t,g⋅m)

for all and . The orbit space of this action is the product and the quotient map is , where is the quotient map of the given action. To see that the set is open, it suffices to check that the open set is saturated with respect to the quotient map , or equivalently that it is -invariant with respect to the action of on the product . For this, let and note, using the equivariance of the vector field , that the curve given by is the maximal integral curve of  starting at . In particular, it is defined for the same times that the integral curve starting at the point is defined. Thus, if then , or equivalently the flow domain  is -invariant. Furthermore, note that . Hence,

 {0}×M/G=(id×π)({0}×M)⊆(id×π)(A)=B

as desired.

Next, note that the map is well-defined by the equivariance of the flow . Furthermore, the map makes the square (2.1) commute by definition. By the characteristic property of the quotient topology, the map is continuous if and only if the map is continuous. Since the diagram (2.1) commutes, we have that . Since is the composition of continuous maps, then the map is continuous.

Now, for every point we have

 Φ(0,π(m))=π(ϕ(0,m))=π(m)

since . Similarly, the second property follows by the corresponding property of the flow

 Φ(t,Φ(s,π(m)))=Φ(t,π(ϕ(s,m)))=π(ϕ(t,ϕ(s,m))) Φ(t,Φ(s,π(m)))=π(ϕ(s+t,m))=Φ(s+t,π(m)).

Hence, the map is the desired abstract flow. ∎

The following is a corollary of Theorem 2.18 and Lemma 2.20 (also see [12, Corollary 2.8]):

###### Corollary 2.21.

Let be a proper -manifold, and let and be two isomorphic equivariant vector fields on . Then the maximal flows of and have the same domain and induce the same abstract flow on the orbit space .

###### Proof.

Let an be the maximal flows of the vector fields and respectively. Let be the family of maps relating the flows (see Theorem 2.18). Thus, for all pairs in the domain of the flow we have

 ϕX(t,m)=Ft(m)⋅ϕY(t,m). (2.2)

In particular, note that any pair in the domain of the flow is in the domain of the flow . Reversing the role of and in Theorem 2.18 gives the opposite inclusion of the flow domains. Hence, the flows and have the same domain.

Now let and be the induced flows on the orbit space of and respectively. Using equality (2.2) and the definition of the induced orbit space flow given in the proof of Lemma 2.20, we have that

 ΦX(t,π(m))=π(ϕX(t,m))=π(Ft(m)⋅ϕY(t,m))=π(ϕY(t,m))=ΦY(t,π(m)).

Hence, the induced flows on the orbit space are equal. ∎

Abstract flows can also have fixed points:

###### Definition 2.22 (fixed point of an abstract flow).

Let be an abstract flow on a topological space . A fixed point of the flow is a point such that for all times with .

###### Remark 2.23.

Let be a proper -manifold and let be the quotient map. Observe that if is an equivariant vector field on , then a point is a relative equilibrium of if and only if the point is a fixed point of the induced abstract flow on the orbit space.

We proceed to describe Krupa’s decomposition following Lerman [12]. We begin by recalling saturations and equivariant extension:

###### Definition 2.24 (saturation).

Given a -manifold and a subset , the saturation of  is the subset of defined by .

Recall that equivariant maps out of regular submanifolds of a proper -manifold have unique equivariant extensions to the saturation of the submanifold, provided some additional hypotheses as in the following standard lemma (see also [5, Lemma 2.10.1]):

###### Lemma 2.25.

Let and be proper -manifolds, let be a Lie subgroup of , let be a -invariant regular submanifold of , and let be a -equivariant map. Suppose that the map given by descends to a diffeomorphism from the associated bundle to the saturation . Then there exists a unique -equivariant extension of the map given by

 εf: G⋅A→N,εf(g⋅a):=g⋅f(a).
###### Proof.

Define the -equivariant map

 G×A→N,(g,a)↦g⋅f(a).

By using the -equivariance of the map , note that this map is -invariant with respect to the action of on . Thus, this map descends to a smooth -equivariant map

 G×KA→N,[g,a]↦g⋅f(a).

Using the diffeomorphism between the associated bundle and the saturation , we obtain the smooth extension . To see it is unique, suppose that is any other -equivariant extension. Then note , and hence . ∎

We recall the definition of a slice:

###### Definition 2.26 (slices).

Given a -manifold , let be the stabilizer of a point . A slice for the action through is a -manifold and a -equivariant embedding such that

1. the point is in the image ;

2. the saturation is open in ;

3. the map

 G×V→G⋅j(V),(g,v)↦g⋅j(v)

descends to a -equivariant diffeomorphism

 G×GmV→G⋅j(V),[g,v]↦g⋅j(v),

where is the associated bundle.

For the sake of conciseness, we often write instead of .

###### Remark 2.27.

It is a classic theorem of Palais [20] that slices exist for points in proper -manifolds (see also [4, Theorem 2.3.3]). In proper -manifolds, it is also possible and convenient to take the slice through a point to be an open ball around the origin of a vector space with a representation of the stabilizer (see, for example, [7, Theorem B.24]).

The following definition will be convenient for the sake of brevity:

###### Definition 2.28 (proper G-manifold with slice).

A proper -manifold with slice is a quintuple consisting of a proper -manifold , a point on the manifold, and a slice  for the action through the point with corresponding -equivariant embedding .

###### Remark 2.29.

Let be a proper -manifold with slice. The following facts will be important:

1. The bundle has typical fiber and is -equivariantly diffeomorphic to the associated bundle (see, for example, [4, Theorem 2.4.1]). In other words, the following diagram, with the canonical maps, commutes:

 \xy(−14,10)∗+G×GmV="1";(12,10)∗+G⋅V="2";(−14,−6)∗G/Gm="3";(12,−6)∗+G⋅m.="4";\ar@−>≅"1";"2";\ar@−>"1";"3";\ar@−>"2";"4";\ar@−>≅"3";"4";\endxy

We think of as a tubular neighborhood of the group orbit and often refer to it as a tube. Thus, the associated bundle serves as a model for the tubular neighborhood , and we will sometimes identify and .

2. Definition 2.26 implies that the tangent space at the point splits in the form

 TmM=Tm(G⋅m)⊕Tmj(V),

while for any point we have

 Tj(v)M=Tj(v)(G⋅j(v))+Tj(v)j(V).

Definition 2.26 also implies that for any point , if a group element is such that then .

###### Remark 2.30.

By the previous remark, we can model tubes generated by slices by considering arbitrary associated bundles of the form , where is a compact Lie subgroup of a Lie group , and is an open ball around the origin in a vector space with a representation of . For such models, note that the point has as stabilizer the Lie subgroup acting on as a subgroup of . Therefore, the -manifold with the -equivariant embedding defined by , is a slice for the action through the point .

###### Remark 2.31.

A proper -manifold with slice gives rise to two action groupoids, namely:

• the action groupoid of the slice;

• the action groupoid of the tube.

Thus, the choice of slice gives rise to two groupoids of equivariant vector fields in the sense of Definition 2.9:

• the groupoid of -equivariant vector fields on the slice ;

• the groupoid of -equivariant vector fields on the tube .

It is a theorem of Lerman that these groupoids are equivalent (see [12, Theorem 1.16]). This theorem was stated using -term chains of topological vector spaces. In Theorem 2.39 we state his result using an equivalent formulation.

Given a proper -manifold with slice, we can use the embedding of the slice to push forward vector fields and infinitesimal gauge transformations onto the image of the slice. We can then extend these uniquely to the tube as in Lemma 2.25. This assembles into a canonical functor as follows:

###### Definition 2.32 (equivariant extension functor).

Let be a proper -manifold with slice. The equivariant extension functor is the functor

 \xy(−25,10)∗+E: X(Gm×V⇉V)="1";(25,10)∗+X(G×G⋅V⇉G⋅V),="2";\ar@−>"1";"2";(−40,0)∗+(X="3";(−20,0)∗+Y)="4";\ar@−>(ψ,X)"3";"4";(5,0)∗+(E(X)="5";(40,0)∗+E(Y)),="6";\ar@−>(E(ψ),E(X))"5";"6";\ar@|−>(−10,0);(−5,0);\endxy

where for any vector field we define

 E(X): G⋅V→T(G⋅V),E(X)(g⋅j(v)):=d(g∘j)X(v),

and for any infinitesimal gauge transformation we define

###### Remark 2.33.

The equivariant extension functor makes use of push-forwards by the slice embedding and of equivariant extension as in Lemma 2.25 at both the object and morphism level. Let be a proper -manifold with slice. Using the notation of Lemma 2.25, the equivariant extension functor satisfies

 E(X)=ε(j∗X)andE(ψ)=ε(j∗ψ)

for any equivariant vector field and any infinitesimal gauge transformation on the slice. Furthermore, note that the image under the functor of the space of -equivariant vector fields on the slice consists of the space of -equivariant vertical vector fields on the bundle . That the image is contained in the space of -equivariant vertical vector fields follows from the definition. That the functor on objects is surjective onto the vertical vector fields can be shown by using the functor of Definition 2.37.

The functor is only part of the equivalence stated in Remark 2.31. For a functor in the opposite direction we first need to obtain a connection via a choice of Lie algebra splitting, as follows:

###### Lemma 2.34.

Let be a Lie group with Lie algebra , let be a Lie subgroup with Lie algebra , and let be a proper -manifold. Then a choice of -equivariant splitting gives rise to a -equivariant connection on the associated bundle .

###### Proof.

We show that the given splitting of the Lie algebra gives rise to a bundle projection from the tangent bundle to the vertical bundle . Here, recall that the vertical bundle is a bundle over the total space of the associated bundle . Thus, we show that the Lie algebra splitting induces a connection .

The Lie algebra splitting gives rise to a -equivariant projection . The projection  in turn gives rise to a principal connection on the principal -bundle , where the subgroup acts on by right-multiplication. For any and , this principal connection is given by

The remaining part of the argument consists of showing that the principal connection induces a connection on the associated bundle . This part of the argument is standard. However, we include an overview here so that we can refer to the construction in the sequel (for more details see, for example, [10, Section 11.8]).

Consider the quotient map and the -equivariant map

 Φ×id: TG×TA→TG×TA,(Φ×id)(X,Y)=(Φ(X),Y).

Since the composition is -invariant with respect to the action of the subgroup on the product , there exists a unique smooth map such that the following diagram commutes:

 \xy(−20,10)∗+TG×TA="1";(15,10)∗+TG×TA="2";(−20,−6)∗+TG×KTA="3";(15,−6)∗+TG×KTA.="4";\ar@−>Φ×id"1";"2";\ar@−>ϖ"2";"4";\ar@−>ϖ"1";"3";\ar@−>˜Φ"3";"4";\endxy (2.3)

The map is idempotent since the map is idempotent. Also, the image of the map is the vertical bundle . Hence, is a projection, so it gives a connection on the associated bundle . Furthermore, the map is -equivariant since the map  is -equivariant and the -action commutes with the quotient map . Hence, the map gives the desired -equivariant connection. ∎

###### Definition 2.35 (connection induced by a splitting).

Let be a proper -manifold with slice and let be a -equivariant splitting. The connection induced by the splitting is the -equivariant connection obtained from Lemma 2.34 by setting , setting , and using the canonical -equivariant diffeomorphism . The vertical projection of vector fields induced by the splitting is the map:

 ν: Γ(T(G⋅V))→Γ(V(G⋅V)),ν(X):=˜Φ∘X.
###### Remark 2.36.

Since the connection of Definition 2.35 is equivariant, the vertical projection of vector fields maps -equivariant vector fields to -equivariant vector fields. Hence, we may also take the vertical projection to be a map . In fact, if is any Lie subgroup of , the vertical projection takes -equivariant vector fields to -equivariant vector fields. Hence, we may also take the vertical projection of vector fields to be a map .

Thus, we obtain the following functor that generalizes Krupa’s decomposition from [11]:

###### Definition 2.37 (projection functor).

Let be a proper -manifold with slice, let be a -equivariant splitting, let be the corresponding -equivariant projection, and let be the vertical projection of equivariant vector fields induced by the splitting (see Definition 2.35). The projection functor corresponding to the Lie algebra splitting is the functor

 \xy(−25,10)∗+P: X(G×G⋅V⇉G⋅V)="1";(25,10)∗+X(Gm×V⇉V),="2";\ar@−>"1";"2";(−45,0)∗+(X="3";(−25,0)∗+Y)="4";\ar@−>(ψ,X)"3";"4";(0,0)∗+(P(X)="5";(35,0)∗+P(Y)),="6";\ar@−>(P(ψ),P(X))"5";"6";\ar@|−>(−15,0);(−10,0);\endxy

where for any vector field we define

 P(X): V→TV,P(X):=j∗(ν(X)),

and for any map we define

 P(ψ): V→gm,P(ψ):=j∗(P∘ψ).
###### Remark 2.38.

Recall that given a smooth embedding we can pull back those vector fields on the target manifold that are tangent to the image of the embedding. Hence, if is a proper -manifold; the vertical vector fields on the bundle can be pulled-back by the embedding . Consequently, the functor  of Definition 2.37 is well-defined on objects.

We can now state the equivalence of groupoids mentioned in Remark 2.31, which is due to Lerman. Instead of the functors of Definitions 2.32 and 2.37, Lerman used an equivalent formulation in terms of -term chain complexes. We state the equivalence using the functor formulation:

###### Theorem 2.39 (Lerman, [12, Theorem 4.3]).

Let be a proper -manifold with slice. The equivariant extension functor see Definition 2.32) and the projection functor corresponding to a choice of -equivariant splitting see Definition 2.37) form an equivalence of categories. In particular, such functors satisfy

 P∘E=idandE∘P≃id.

For a given equivariant vector field on the tube , the natural isomorphism is of the form

 αX=(ψX,E(P(X))),

where is an infinitesimal gauge transformation taking values in the complement . Thus, the map is such that

 X=E(P(X))+ψXG⋅V,

where is the vector field induced by the map .

###### Remark 2.40.

Lerman introduced this approach to Krupa’s decomposition to quantify the result of the choices in slice and projection. The choice in slice is adressed as follows. Let be a proper -manifold and a point in . If and are two slices for the action through the point , then the corresponding groupoids and of -equivariant vector fields are isomorphic groupoids [12, Lemma 3.21]. After perhaps shrinking the slices, the isomorphism is induced by a -equivariant diffeomorphism between the slices. The choice in projection, or equivalently the choice of Lie algebra splitting, is addressed as follows. Given a proper -manifold with slice , and two choices of -equivariant splittings

 g=gm⊕q1=gm⊕q2,

the corresponding projection functors

 P1,P2: X(G×G⋅V⇉G⋅V)→X(Gm×V⇉G×V)

are naturally isomorphic [12, Lemma 3.17].

As may be expected, the functors we have introduced preserve relative equilibria. We prepare for the proof of this fact via Lemmas 2.41, 2.42, and 2.43.

###### Lemma 2.41.

Let and be proper -manifolds and let be a -equivariant diffeomorphism. Suppose that and are -related equivariant vector fields on and respectively. Then a point is a relative equilibrium of the vector field if and only if the point is a relative equilibrium of the vector field . Thus, pullbacks and pushforwards of vector fields by equivariant diffeomorphisms preserve relative equilibria.

###### Proof.

The verification is a straightforward computation using the equation . First, suppose is a -relative equilibrium of the vector field . Then

 Y(f(m))=(df)m(X(m))∈(df)m(Tm(G⋅m))=Tf(m)(G⋅f(m)),

where follows by the equivariance of the diffeomorphism . Thus, the point is a -relative equilibrium of the vector field . The converse is completely analogous. ∎

###### Lemma 2.42.

Let be a proper -manifold, let be a Lie subgroup of , and let  be a -invariant regular submanifold of satisfying the hypotheses of Lemma 2.25. Suppose that  is a -equivariant vector field on and that the point is a -relative equilibrium of the vector field . Then the point  is a -relative equilibrium of the equivariant extension  of . That is, equivariant extension preserves relative equilibria.

###### Proof.

This is essentially a corollary of Lemma 2.41. Let be the inclusion of the submanifold . Note that the tube is a -manifold and that the inclusion is a -equivariant diffeomorphism onto its image . Observe that the vector fields and are -related; in fact, restricts to on . Thus, by Lemma 2.41, we know that is a -relative equilibrium of the vector field ; that is, . Since is a regular submanifold, the tangent space is contained in the tangent space . Hence, the point  is a -relative equilibrium of . ∎

###### Lemma 2.43.

Let be a Lie group, let be a Lie subgroup, let be a proper -manifold, and let be a -equivariant splitting. Let be the vertical projection induced by the splitting Definition 2.35), and let  be a -equivariant vector field on the associated bundle . Then if the point is a -relative equilibrium of the vector field , it is also a -relative equilibrium of the vertical projection . That is, the vertical projection preserves relative equilibria.

###### Proof.

Consider the quotient maps:

 π: G×A→G×KA,ϖ: TG×TA→TG×KTA.

Let be an equivariant vector field on the associated bundle and suppose that the point is such that the point is a relative equilibrium of the vector field . Let and be the connections induced by the splitting of the Lie algebra (see Definition 2.35 and the proof of Lemma 2.34), and let the map

 ν: Γ(T(G×KA))G→Γ(V(G×KA))G

be the vertical projection of equivariant vector fields with respect to this connection (Definition 2.35). We want to show that the point is a -relative equilibrium of the vector field ; that is, we want to show that

 ν(X)(m)∈Tm(G⋅m).

Since the action of commutes with the quotient maps and , observe that

 ϖ(Tp(G⋅p))=Tm(G⋅m). (2.4)

Furthermore, using that , it is clear that

 (Φ×id)(Tp(G⋅p))⊆Tp(G