Automorphisms of generic gradient vector fields with prescribed finite symmetries
aa r X i v : . [ m a t h . DG ] N ov AUTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDSWITH PRESCRIBED FINITE SYMMETRIES
IGNASI MUNDET I RIERA
Abstract.
Let M be a compact and connected smooth manifold endowed with asmooth action of a finite group Γ, and let f be a Γ-invariant Morse function on M . Weprove that the space of Γ-invariant Riemannian metrics on M contains a residual subset M et f with the following property. Let g ∈ M et f and let ∇ g f be the gradient vectorfield of f with respect to g . For any diffeomorphism φ ∈ Diff( M ) preserving ∇ g f thereexists some t ∈ R and some γ ∈ Γ such that for every x ∈ M we have φ ( x ) = γ Φ gt ( x ),where Φ gt is the time- t flow of the vector field ∇ g f . Contents
1. Introduction 12. Proof of Theorem 1.1 for dim M = 1 53. Equivariant Sternberg’s linearisation theorem for families 94. The space of metrics M et S g ( p ), the distributions A g ( p ), and the sets F g ( p ) 146. The space of metrics M et ,K M >
Introduction
Define the automorphism group of a vector field on a smooth manifold to be the groupof diffeomorphisms of the manifold preserving the vector field. A natural question is howsmall the automorphism group of a vector field can be. Suppose that we only considervector fields which are invariant under a fixed finite group action on the manifold. Inthis situation, the automorphism group of the vector field always includes the action ofthe finite group and the flow of the vector field. Our main result implies that if themanifold is compact and connected then the set of invariant gradient vector fields whose
Date : October 30, 2017.2010
Mathematics Subject Classification. automorphism group contains nothing more than this is residual in the set of all invariantgradient vector fields. (Recall that a residual subset is a countable intersection of denseopen subsets. Since the space of smooth invariant gradient vector fields is Baire , any ofits residual subsets is dense.)Let us explain in more concrete terms our motivation and main result.Take M to be a smooth (= C ∞ ) manifold, and denote by X ( M ) the vector space ofsmooth vector fields on M , endowed with the C ∞ topology. Denote the automorphismgroup of a vector field X ∈ X ( M ) byAut( X ) = { φ ∈ Diff( M ) | φ ∗ X = X } . If X = 0 then Aut( X ) contains a central subgroup isomorphic to R , namely the flowgenerated by X . Denote by Aut( X ) / R the quotient of Aut( X ) be this subgroup.Suppose that M is endowed with a smooth and effective action of a finite group Γ. Let X ( M ) Γ ⊂ X ( M ) be the space of Γ-invariant vector fields on M . F.J. Turiel and A. Viruelproved recently in [22] that there exists some X ∈ X ( M ) Γ such that Aut( X ) / R ≃ Γ.The vector field X is given explicitly in [22] as a gradient vector field for a carefullyconstructed Morse function and a suitable Riemannian metric. One may wonder, inview of that result, whether the set of X ∈ X ( M ) Γ satisfying Aut( X ) / R ≃ Γ is genericin some sense, at least if one restricts to some particular family of vector fields (such asgradient vector fields, for example). Here we give an affirmative answer to this questionassuming M is compact and connected.Suppose, for the rest of the paper, that M is compact and connected. For any function f ∈ C ∞ ( M ) and any Riemannian metric g on M we denote by ∇ g f ∈ X ( M ) the gradientof f with respect to g , defined by the condition that g ( ∇ g f, v ) = df ( v ) for every v ∈ T M .The following is our main result.
Theorem 1.1.
Let M et ⊂ C ∞ ( M, S T ∗ M ) denote the space of Riemannian metrics on M , with the C ∞ topology, and let M et Γ ⊂ M et denote the space of Γ -invariant metrics.Let f be a Γ -invariant Morse function on M . There exists a residual subset M et f ⊂ M et Γ such that any g ∈ M et f satisfies Aut( ∇ g f ) / R ≃ Γ . By a result of Wasserman [24, Lemma 4.8], the space of Γ-invariant Morse functionson M is open and dense within the space of all Γ-invariant smooth functions on M (for openness see the comments before Lemma 4.8 in [24]). Combining this result withTheorem 1.1 it follows that if G ( M ) Γ ⊂ X ( M ) Γ denotes the set of Γ-invariant gradientvector fields, then the set of X ∈ G ( M ) Γ satisfying Aut( X ) / R ≃ Γ contains a residualsubset in G ( M ) Γ .Probably Theorem 1.1 can be proved as well for most proper Γ-invariant Morse func-tions on open manifolds endowed with an smooth effective action of a finite group.However, there are exceptions: if M = R n with n >
1, and Γ is a finite group actinglinearly on M preserving the standard euclidean norm, then f : M → R , f ( v ) = k v k ,is a Γ-invariant Morse function, and for any Γ-invariant Riemannian metric g on M ,Aut( ∇ g f ) / R is bigger than Γ. This follows from a result of Sternberg, see Theorem 3.1below. See e.g. [9, Chap. 2, Theorem 4.4] and the comments afterwards.
UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 3
For the case dim M = 1 (i.e., when M is the circle) we prove a stronger form ofTheorem 1.1, where residual is replaced by open and dense, see Theorem 2.1. It is notinconceivable that this can be done in all dimensions: while the author has not managedto do so, he does not have any reason to suspect that it might be false. In fact, a theoremof Palis and Yoccoz answering an analogous question (in the non-equivariant setting),with the set of gradient vector fields X ( M ) replaced by a certain set of diffeomorphismsmay suggest that it is true. To be precise, let A ( M ) ⊂ Diff( M ) be the (open) set ofAxiom A diffeomorphisms satisfying the transversality condition and having a sink or asource. Palis and Yoccoz prove in [14] that the set of diffeomorphisms with the smallestpossible centralizer contains a C ∞ open and dense subset of A ( M ). Note that the set A ( M ) includes Morse–Smale diffeomorphisms, which are analogues for diffeomorphismsof gradient vector fields. However, if one considers the same question for the entirediffeomorphism group endowed with the C topology then openness never holds, see [4].Before explaining the main ideas in the proof of Theorem 1.1, let us discuss somerelated problems. A natural question is whether the property proved in Theorem 1.1 istrue when replacing the space of invariant gradient vector fields by the entire space ofinvariant vector fields. Problem A.
Does the set of X ∈ X ( M ) Γ satisfying Aut( X ) / R ≃ Γ contain a residualsubset of X ( M ) Γ ?Problem A includes Theorem 1.1 as a particular case, since G ( M ) Γ is open in X ( M ) Γ .Note that the case Γ = { } of Problem A (or even Theorem 1.1) is far from being trivial.Define the centralizer Z ( X ) of a vector field X on M to be the group of diffeomorphismsof M that send orbits of X onto orbits of X . For any X , Aut( X ) is a subgroup of Z ( X ),and one may try to explore analogues of Theorem 1.1 and Problem A for Z ( X ). However,even the right question to ask is not clear in this situation. P.R. Sad [16] studied thecase Γ = { } . His main result is that for a compact M there is an open and dense subset A ′ of the set of Morse–Smale vector fields A ⊂ X ( M ) such that for any X ∈ A ′ there isa neighborhood V ⊂ Diff( M ) of the identity with the property that any φ ∈ V ∩ Z ( X )preserves the orbits of X . Unfortunately the restriction to a neighborhood of the identityin Diff( M ) can not be removed, as Sad shows with an example.It is natural to consider analogues of the previous problems replacing vector fields bydiffeomorphisms. Define the automorphism group of a diffeomorphism φ ∈ Diff( M ) tobe its centralizer, i.e., Aut( φ ) = { ψ ∈ Diff( M ) | φψ = ψφ } . Then h φ i = { φ k | k ∈ Z } isa central subgroup of Aut( φ ). LetDiff Γ ( M ) = { φ ∈ Diff( M ) | φ commutes with the action of Γ } . Problem B.
Does the set of φ ∈ Diff Γ ( M ) such that Aut( φ ) / h φ i ≃ Γ contain a residualsubset of Diff Γ ( M )?Of course, a positive answer to Problem B does not imply a positive answer to ProblemA, since a diffeomorphism φ such that Aut( φ ) / h φ i = Γ can not possibly belong to theflow of a vector field (for otherwise Aut( φ ) should contain a subgroup isomorphic to R ).One may consider restricted versions of Problem B involving particular diffeomor-phisms, for example, equivariant Morse–Smale diffeomorphisms [8]. These are very par-ticular diffeomorphisms, but Problem B is already substantially nontrivial for them (evenin the case Γ = { } , see below). IGNASI MUNDET I RIERA
Problems A and B admit variations in which the regularity of the vector fields or thediffeomorphisms is relaxed from C ∞ to C r for finite r . One can also consider strongerquestions replacing residual by open and dense or weaker ones replacing residual by dense .The case Γ = { } of Problem B is a famous question of Smale. It appeared for the firsttime in [17, Part IV, Problem (1.1)], in more elaborate form in [18], and it was includedin his list of 18 problems for the present century [19]. It was solved for Morse–Smale C -diffeomorphisms by Togawa [21] and very recently for arbitrary C -diffeomorphisms by C.Bonatti, S. Crovisier, A. Wilkinson in [2] (see the survey [3] for further references). Theanalogous problem for higher regularity diffeomorphisms is open at present, althoughthere are by now plenty of partial results: see e.g. [11] for the case of the circle, [14] forelements in the set A ( M ) defined above, and [15] for Anosov diffeomorphisms of tori.Theorem 1.1 may be compared to similar results for other types of tensors. Forexample, it has been proved in [12] that on a compact manifold the set of metrics offixed signature with trivial isometry group is open and dense in the space of all suchmetrics (see also [7] for an infinitesimal version of this with the compactness conditionremoved).1.1. Main ideas of the proof.
To prove Theorem 1.1 we treat separately the casesdim M = 1 and dim M >
1. The case dim M = 1 is addressed in Section 2, using ratherad hoc methods. An interesting ingredient is an invariant of vector fields which, whennonzero, distinguishes changes of orientation, and which plays an important role in theclassification up to diffeomorphisms of vector fields on S with nondegenerate zeroes.The main ingredient in the case dim M >
1, common to other papers addressingsimilar problems, is a theorem of Sternberg [20] on linearisation of vector fields nearsinks and sources, assuming there are no resonances. The use of this result in this kindof problems goes back to work of Kopell [11], and appears in papers of Anderson [1] andPalis and Yoccoz [14] among others. To apply this theorem in our situation we needto generalize it to the equivariant setting under the presence of finite symmetries. Thisposes some difficulties. For example, in the equivariant case we can not suppose that theeigenvalues of the linearisation of a generic vector field at fixed points are all different:high multiplicities can not be avoided; in particular, the centraliser of the linearisationis not necessarily abelian (both Anderson [1] and Palis–Yoccoz [14] restrict themselvesto the case in which the eigenvalues are different). This is relevant for example whenextending the version of Sternberg’s theorem for families proved by Anderson to theequivariant setting (see Section 3 for details on this).We close this subsection with a more concrete description of the proof of the casedim
M >
1. Suppose a Γ-invariant Morse function f has been chosen. The set of metrics M et f is defined as the intersection of a set of invariant metrics, M et , and a countablesequence of subsets { M et ,K } K ∈ N . Each of these sets is open and dense in M et Γ .The metrics g ∈ M et , defined in Subsection 4.3, have two properties: (1) the eigenval-ues of the differential of ∇ g f at each critical point are as much different among themselvesas they can be (in particular, the collection of eigenvalues at two critical points coincideif and only if the two points belong to the same Γ-orbit), and (2) there are no resonancesamong eigenvalues at any critical point. The second property allows us to use Stern-berg’s theorem on linearisation on neighborhoods of sinks and sources, and a theorem of UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 5
Kopell which limits enormously the automorphisms of the gradient vector field restrictedto (un)stable manifolds of sinks/sources.The metrics g ∈ M et ,K , defined in Subsection 6, have the following property. Supposethat p is a sink and W sg ( p ) is the stable manifold of p for ∇ g f , and that q is a sourceand W ug ( q ) is its unstable manifold for ∇ g f . If W sg ( p ) ∩ W ug ( q ) is nonempty, then anyautomorphism of ∇ g f | W sg ( p ) whose derivative at p is at distance ≤ K from the identityand which matches on W sg ( p ) ∩ W ug ( q ) with an automorphism of ∇ g f | W ug ( q ) is at distance < K − from an automorphism coming from the action of Γ and the flow of ∇ g f .After defining these sets of metrics, in Section 7 we prove Theorem 1.1 for manifoldsof dimension greater than one, showing that if g ∈ M et f then Aut( ∇ g f ) / R ≃ Γ.The paper concludes with two appendices. The first one gives the proof of a technicalresult on the variation of the gradient flow of ∇ g f with respect to variations of g , andthe second one contains a glossary of the notation used to address the case dim M >
Acknowledgement.
I am very grateful to the referee for pointing out a substantialsimplification in the proof of the main theorem, which was originally much longer andmore involved, and for detecting a number of mistakes and suggesting improvements.2.
Proof of Theorem 1.1 for dim M = 1In this section we prove a strengthening of the case dim M = 1 of Theorem 1.1. Moreconcretely, in Subsection 2.2 below we prove the following. Theorem 2.1.
Suppose that a finite group Γ acts smoothly and effectively on S . Let f be a Γ -equivariant Morse function on S . Let M et Γ denote the set of Γ -invariantRiemannian metrics on S , endowed with the C ∞ topology. There exists a dense andopen subset M et f ⊂ M et Γ such that for every g ∈ M et f we have Aut( ∇ g f ) / R ≃ Γ . Classifying nondegenerate vector fields on the circle.
To prove Theorem 2.1we will need, in the case when Γ is generated by a rotation, an invariant of nondegeneratevector fields on the circle that detects change of orientations. This invariant is one ofthe ingredients of the classification of nondegenerate vector fields on the circle up toorientation preserving diffeomorphism. Detailed expositions of this classification (in thebroader context of vector fields with zeroes of finite order) have appeared in [6, 10]. Herewe briefly explain the main ideas of this result, focusing on the definition of the invariant,both for completeness and to set the notation for later use.For any t ∈ R and vector field X we denote by Φ X t ∈ Aut( X ) the flow of X at time t .We first consider the local classification of vector fields with a nondegenerate zero.For any nonzero real number λ we denote by F λ the set of germs of vector fields ona neighborhood of 0 in R of the form h ∂ x , where h (0) = 0 and h ′ (0) = λ and x isthe standard coordinate in R . Let G denote the group of germs of diffeomorphisms ofneighborhoods of 0 in R . For any X ∈ F λ we denote by Aut( X ) the group of all φ ∈ G such that φ ∗ X = X . For example, Φ X t ∈ Aut( X ) for every t . The proof of the next lemmafollows from a straightforward computation and Cauchy’s theorem on ODE’s. Lemma 2.2.
Let λ, µ be nonzero real numbers.
IGNASI MUNDET I RIERA (1)
Given X ∈ F λ and Y ∈ F µ there exists some φ ∈ G satisfying φ ∗ X = Y if and onlyif λ = µ . (2) For any λ and X ∈ F λ the map D : Aut( X ) → R ∗ sending φ to φ ′ (0) is anisomorphism of groups. Furthermore, D Φ X t = e λt . We mention in passing that to prove the case dim
M > S with R / π Z , so vector fields on S can be written as X = h ∂ x where h is a 2 π -periodic smooth function. We say that X is nondegenerate if h ( y ) = 0implies h ′ ( y ) = 0 ( h ′ ( y ) can be identified with the derivative of X at y ∈ h − (0)).An immediate consequence is that h contains finitely many zeroes in [0 , π ). Anotherconsequence is that h changes sign when crossing any zero of h , and this implies that h − (0) contains an even number of elements in [0 , π ). To classify nondegenerate vectorfields on the circle we will associate to them the number of zeroes, their derivatives atthe zeroes (up to cyclic order), and a global invariant denoted by χ .To define χ suppose first of all that h has no zeroes. Denoting by Φ X t the flow of X seen as a vector field on R , there is a unique real number t such that Φ X t ( y ) = y + 2 π forevery y ∈ R . Then we set χ ( X ) := t. Now suppose that h vanishes somewhere, and write its zeroes contained in [0 , π ) as0 ≤ z < z < · · · < z r < π. We extend this finite list to an infinite sequence by setting z i +2 r = z i for every integer i .Below, we implicitly consider similar periodic extensions for all objects that we are goingto associate to the zeroes z i . By (2) in Lemma 2.2, for every i there exists a connectedneighborhood U i of z i , disjoint from z i − and z i +1 , and a unique smooth involution σ i : U i → U i such that(1) σ i ( z i ) = z i , σ ′ i ( z i ) = − , ( σ i ) ∗ X = X . Choose for every i some t + i > z i contained in U i and define t − i = σ i ( t + i ). Then we have t + i , t − i +1 ∈ ( z i , z i +1 ), so there is a unique real number ρ i such that t − i +1 = Φ X ρ i ( t + i ) . Note that ρ i has the same sign as h ′ ( z i ). Now we define(2) χ ( X ) := r X i =1 ρ i . UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 7
Lemma 2.3.
The number χ ( X ) only depends on X , and not on the choices of t ± i . Further-more, endowing the set of generic vector fields with the C ∞ topology the map X χ ( X ) is continuous.Proof. We first prove that χ ( X ) does not depend on the choices of t ± i . If for any i wereplace t ± i by ( t ′ i ) ± , then the requirement that ( t ′ i ) − = σ i (( t ′ i ) + ) implies that ( t ′ i ) ± =Φ X ± δ ( t ± i ) for some δ , so ρ i gets replaced by ρ i − δ and ρ i − gets replaced by ρ i − + δ , andhence (2) remains unchanged.To prove that χ ( X ) depends continuously on X we first observe that any other vectorfield sufficiently close to X is also generic and has vanishing locus close to that of X .Hence, once we have fixed the intervals U i and points t + i above, there is a neighborhood V of X in the space of all vector fields on the circle such that if Y ∈ V and we write Y = k ∂ x then k − (0) ⊂ S i U i , each U i contains a unique zero w i of k , and w i < t + i . So itsuffices to prove that given δ >
0, choosing V small enough, the involution σ Y i satisfying(1) with X resp. z i replaced by Y resp. w i has the property that σ Y i ( t + i ) is well definedand at distance < δ from σ i ( t + i ).The previous property will follow if we prove that σ i depends continuously on X .This is a local question, so let us assume that X is a vector field defined on an openinterval 0 ∈ I ⊂ R with X = g ∂ x and satisfying g − (0) = { } and g ′ (0) = 0; byLemma 2.2 there is an open interval 0 ∈ J ⊂ R and a smooth embedding φ : J → I such that φ ∗ ( λx ∂ x ) = X for some nonzero real λ . It is easy to check that both φ and λ depend continuously on X . Take U = φ ( J ∩ − J ). The map σ : U → U defined as σ ( x ) = φ ( − φ − ( x )) is a smooth involution of U and it satisfies σ ∗ X = X . By the previousobservations it is clear that σ depends continuously on X . (cid:3) This is the classification theorem of nondegenerate vector fields on S : Theorem 2.4.
Given two vector fields X and Y on the circle, there exists an orientationpreserving diffeomorphism φ ∈ Diff + ( S ) satisfying φ ∗ X = Y if and only if X and Y havethe same number of zeroes, the collection of derivatives at the zeroes of X and Y , travellingalong S counterclockwise, coincide up to a cyclic permutation, and χ ( X ) = χ ( Y ) . We are not going to prove the previous theorem. In fact we will only use the ”only if”part of it, which is rather obvious from the definitions; the proof of the ”if” part is aneasy exercise using Lemma 2.2. See [6, 10] for detailed proofs of a more general result.We close this subsection with another result that will be used in the proof of Theorem2.1. Suppose that h : R → R is a smooth function such that h (0) = h (1) = 0, and that h does not vanish on the open interval (0 , X = h ∂ x . The next lemma followseasily from Cauchy’s theorem on ODE’s. Lemma 2.5.
Any diffeomorphism φ : (0 , → (0 , satisfying φ ∗ X = X is equal to Φ X t for some t ∈ R . In particular if a diffeomorphism φ : (0 , → (0 , satisfying φ ∗ X = X is the identity on an open subset of (0 , then φ is the identity on the entire (0 , . Proof of Theorem 2.1.
Let Γ be a finite group acting smoothly and effectively on S , and let f : S → R be a Γ-invariant Morse function. Let Crit( f ) be the set of criticalpoints of f . Any Γ-invariant Riemannian metric in M et Γ is isometric to the round circle { x + y = r } IGNASI MUNDET I RIERA for some r >
0, and this allows to identify the action of Γ on S with the action of acyclic or a dihedral group. We treat separately the two possibilities.2.2.1. Dihedral groups.
Suppose first that Γ is dihedral. Then Γ contains elements thatreverse the orientation. Let p ∈ S be a fixed point on an orientation reversing element ofΓ, and let Γ ⊂ Γ be the subgroup of the elements which act preserving the orientation.Since [Γ : Γ ] = 2, we have Γ p = Γ p . On the other hand, p is necessarily a criticalpoint of f , because f is Γ-invariant.Let M et f be the set of metrics g ∈ M et Γ satisfying:(3) if x, y ∈ Crit( f ) and D ∇ g f ( x ) = D ∇ g f ( y ) then Γ x = Γ y .It is clear that M et f is open and dense in M et Γ . Now suppose that g ∈ M et f and let X = ∇ g f . To prove that Aut( X ) / R ≃ Γ we consider an arbitrary φ ∈ Aut( X ) and showthat composing φ with the action of suitably chosen elements of Γ and with the flow Φ X t for some t we obtain the identity.Let φ ∈ Aut( ∇ g f ). Composing φ with the action of some γ ∈ Γ we may assume that φ is orientation preserving. By (3) we have φ ( p ) ∈ Γ p . Since Γ p = Γ p , up to composing φ with the action of some element of Γ we can assume that φ preserves the orientationand fixes p . This implies that φ fixes all critical points of f .Let us label counterclockwise the critical points of f as p , p , . . . , p r . By Lemma 2.2,up to composing φ with Φ X t for some choice of t we may assume that φ is the identityon a neighborhood of p . This implies that φ is the identity on the entire circle. Indeed,by Lemma 2.5, φ is the identity on the arc from p to p , so by Lemma 2.2 φ is theidentity on a neighborhood of p . We next apply Lemma 2.5 to the arc from p to p and conclude that the restriction of φ to this arc is equal to the identity. An so on, untilwe have traveled around the entire circle.2.2.2. Cyclic groups.
Suppose that Γ is a cyclic group. The only case in which Γ cancontain orientation reversing elements is that in which Γ consists of two elements, thenontrivial one being an orientation reversing involution of S . This situation can beaddressed with the arguments of the previous case, so let us assume here that all elementsof Γ preserve the orientation. Then we define M et f to be the set of metrics g ∈ M et Γ satisfying property (3) above and χ ( ∇ g f ) = 0.We claim that M et f is open and dense in M et Γ . Since the set of metrics g ∈ M et Γ satisfying property (3) is open and dense, to see that M et f is dense it suffices to observethat if for some choice of g we have χ ( ∇ g f ) = 0 then slightly modifying g away from thecritical points we may force χ to take a nonzero value; furthermore, the modification of g can be made Γ-invariant because Γ is generated by a rotation (note that, in contrast, ifΓ is a dihedral group then for any Γ-invariant metric g we have χ ( ∇ g f ) = 0). Opennessof M et f follows from the second statement in Lemma 2.3.Let g ∈ M et f , let X = ∇ g f , and let φ ∈ Aut( X ). We claim that φ is orienta-tion preserving. Indeed, for any orientation reversing diffeomorphism ψ of S we have χ ( ψ ∗ X ) = − χ ( X ) and since χ ( X ) = 0, we can not possibly have ψ ∗ X = X . Let p be anycritical point of f . By (3) we have φ ( p ) ∈ Γ p , so up to composing φ with the action ofsome element of Γ we can assume that φ ( p ) = p . Then, since φ preserves the orientation, UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 9 it fixes all the other critical points, and the argument is concluded as in the case ofdihedral groups. Hence the proof of Theorem 2.1 is now complete.In the remainder of the paper we are going to assume that dim
M >
Equivariant Sternberg’s linearisation theorem for families
The following is Sternberg’s linearisation theorem [20, Theorem 4], which extends tothe smooth setting an analytic result proved by Poincar´e in his thesis:
Theorem 3.1 (Sternberg) . Let ∈ U ⊂ R n be an open set and let X : U → R n be asmooth vector field satisfying X (0) = 0 . Suppose that the derivative D X (0) diagonalisesand has (possibly complex) eigenvalues λ , . . . , λ n , repeated with multiplicity. Supposethat each λ i has negative real part, and that (4) λ i = n X j =1 α j λ j , for any i , and any α , . . . , α n ∈ Z ≥ satisfying X α j ≥ . Then there exists open sets ∈ U ′ ⊂ R n and ∈ U ′′ ⊂ U , and a diffeomorphism φ : U ′′ → U ′ , such that Dφ (0) = Id and φ ◦ X ◦ φ − = D X (0) . Actually [20, Theorem 4] states that φ can be chosen to be C k for every finite and bigenough k . The fact that φ can be assumed to be C ∞ follows from [11, Theorem 6].Sternberg proved in [20] an analogous theorem for local diffeomorphisms of R n . Later,Anderson proved [1, §
2, Lemma] a parametric version of Sternberg’s theorem for diffeo-morphisms, which can be translated, using the arguments in [20, § D ⊂ R n be an open diskcentered at 0, and let ∆ ⊂ D be a smaller concentric disk. Let r be a natural number.For any smooth map X : D → R n define k X k ∆ ,r = sup x ∈ ∆ k D r X ( x ) k , where k D r X ( x ) k denotes the sum of the norms of all partial derivatives of X at x of degree ≤ r . Thisdefines a (non separated!) topology on Map ( D, R n ), the set of all smooth maps D → R n fixing 0, and we denote by Map ( D, R n ) ∆ ,r the resulting topological space. This is theanalogue of Anderson’s theorem for vector fields: Theorem 3.2.
Let L : R n → R n be a linear map which diagonalises with eigenvalues λ , . . . , λ n satisfying (4). Assume that each λ i has negative real part, and that λ i = λ j for i = j . There exists a neighborhood N of L in Map ( D, R n ) ∆ ,r +1 and a continuousmap Φ : N → Map ( D, R n ) ∆ ,r such that: (1) for every X ∈ N , D Φ( X )(0) = Id , so Φ( X ) gives a diffeomorphism U X → U ′ X between neighborhoods of , (2) for every X ∈ N , Φ( X ) ◦ X ◦ Φ( X ) − : U ′ X → R n is equal to D X (0) . We will need an analogue of Theorem 3.2 in an equivariant setting. However, as wasmentioned in the introduction, the presence of symmetries usually forces eigenvalues tohave high multiplicity, and consequently the hypothesis in Theorem 3.2 will most of thetimes not hold.
Now the (only) reason why Anderson assumes the eigenvalues λ , . . . , λ n to be pair-wise distinct is that he needs to be able to diagonalize the linear maps close to L ina continuous way. To state this more precisely, let GL ∗ ( R , n ) ⊂ GL( n, R ) denote theopen and dense set of linear automorphisms of R n all of whose eigenvalues are distinct.Anderson uses the following elementary lemma. Lemma 3.3.
Any L ∈ GL ∗ ( n, R ) admits a neighborhood U ⊂ GL ∗ ( n, R ) and smoothmaps f , . . . , f n : U → C n so that for any L ′ ∈ U the vectors f ( L ′ ) , . . . , f n ( L ′ ) form abasis of C n with respect to which L ′ diagonalizes. So to obtain an equivariant analogue of Theorem 3.2 it suffices to define some openand dense subset of the set of equivariant automorphisms of a vector space enjoying thesame property as GL ∗ ( n, R ). This is the purpose of the following lemma, which alsoproves a property on centralizers that will be used later in the paper.Suppose that V is an n -dimensional real vector space, and that a finite group G actslinearly on V . Denote the centralizer of any Λ ∈ Aut( V ) by Z (Λ) = { Λ ′ ∈ Aut( V ) | ΛΛ ′ = Λ ′ Λ } . Let Aut G ( V ) denote the Lie group of automorphisms of V commuting with the G action.Define Aut ∗ G ( V ) to be the set of all Λ ∈ Aut G ( V ) such that for any λ ∈ C the ( G -invariant) subspace Ker(Λ − λ Id) ⊂ V is irreducible as a representation of G . Givena basis a , . . . , a n ∈ V ⊗ C we denote by ( a , . . . , a n ) : C n → V ⊗ C the isomorphism( λ , . . . , λ n ) P λ i a i . Lemma 3.4.
The subset
Aut ∗ G ( V ) is open and dense in Aut G ( V ) . Any Λ ∈ Aut ∗ G ( V ) has a neighborhood U ⊂ Aut ∗ G ( V ) with smooth maps f , . . . , f n : U → V ⊗ C so that forany Λ ′ ∈ U the vectors f (Λ ′ ) , . . . , f n (Λ ′ ) form a basis of V ⊗ C with respect to which Λ ′ diagonalizes, and conjugation by ( f ′ , . . . , f ′ n )( f , . . . , f n ) − gives an isomorphism Z (Λ) ≃ −→ Z (Λ ′ ) . Proof.
Denote by b G the set of irreducible characters of G . For any ξ ∈ b G let V ξ bea G -representation with character ξ . As a G -representation, we may identify V with L ξ ∈ b G V ξ ⊗ E ξ , where each E ξ is a vector space with trivial G -action. By Schur’s lemmathe space of G -equivariant endomorphisms of V isEnd G ( V ) = M ξ ∈ b G End E ξ . An endomorphism Λ = (Λ ξ ) ξ (where Λ ξ ∈ End E ξ for each ξ ) belongs to Aut G ( V ) exactlywhen Q ξ det Λ ξ = 0, and it belongs to Aut ∗ G ( V ) if and only if, additionally, no root of thepolynomial Q ξ det(Λ ξ − x Id E ξ ) ∈ R [ x ] has multiplicity bigger than one. This conditionimplies that Λ ξ ∈ GL ∗ ( E ξ ) for each ξ . Applying Lemma 3.3 to each Λ ξ we deduce theexistence of a neighborhood U ⊂ Aut ∗ G ( V ) of Λ and smooth maps f , . . . , f n : U → V ⊗ C and λ , . . . , λ n : U → C so that for any Λ ′ ∈ U we have Λ ′ ( f j (Λ ′ )) = λ j (Λ ′ ) f j (Λ ′ ) forevery j . For any Λ ′ ∈ U we can identify Z (Λ ′ ) with the subgroup of Aut( V ) preservingthe subspace of V ⊗ C spanned by { f j (Λ ′ ) | λ j (Λ ′ ) = λ } for each λ . Shrinking U ifnecessary we may assume that for any i, j and any Λ ′ ∈ U we have λ i (Λ ′ ) = λ j (Λ ′ ) ⇐⇒ λ i (Λ) = λ j (Λ) , UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 11 so conjugation by ( f ′ , . . . , f ′ n )( f , . . . , f n ) − gives an isomorphism Z (Λ) ≃ −→ Z (Λ ′ ). (cid:3) Take some G -invariant Euclidean metric on V , let D ⊂ V be an open disk centeredat 0, and let ∆ ⊂ D be a smaller concentric disk. Let r be a natural number. Forany smooth map X : D → V define k X k ∆ ,r = sup x ∈ ∆ k D r X ( x ) k as before. This de-fines a topology on Map G, ( D, V ), the set of all G -equivariant smooth maps D → V fixing 0. Let Map G, ( D, V ) ∆ ,r be the resulting topological space. Define analogouslyMap ( D, V ) ∆ ,r by dropping the equivariance condition. Combining the previous lemmawith the arguments in [1, §
2, Lemma] and [20, §
6] we obtain the following.
Theorem 3.5.
Let L ∈ Aut ∗ G ( V ) have eigenvalues λ , . . . , λ n satisfying (4) and supposethat each λ i has negative real part. There is a neighborhood N of L in Map G, ( D, V ) ∆ ,r +1 and a continuous map Φ : N → Map G, ( D, V ) ∆ ,r such that: (1) for every X ∈ N , D Φ( X )(0) = Id , so Φ( X ) gives a diffeomorphism U X → U ′ X between neighborhoods of , (2) for every X ∈ N , Φ( X ) ◦ X ◦ Φ( X ) − : U ′ X → V is equal to D X (0) . The only part in the statement of Theorem 3.5 that does not follow immediately is thefact that the conjugating map Φ may be chosen to take values in Map G, ( D, V ) ∆ ,r . Stern-berg’s argument provides a (continuous, by Anderson) map Φ : N → Map ( D, V ) ∆ ,r ,satisfying (i) D Φ ( X )(0) = Id and (ii) Φ ( X ) ◦ X ◦ Φ ( X ) − = D X (0) (in a neighborhoodof 0), but Φ ( X ) is not necessarily equivariant. Now, equality (2) is equivalent to(5) Φ ( X ) ◦ X = D X (0) ◦ Φ ( X ) , so setting Φ( X )( x ) = 1 | G | X g ∈ G g Φ ( X )( g − x ) ∈ V for every x ∈ D , we have Φ( X ) ∈ Map G, ( D, V ) ∆ ,r , and equation (5) immediately givesΦ( X ) ◦ X = D X (0) ◦ Φ( X ). Trivially we also have D Φ( X )(0) = Id for every X , andΦ( X ) : D → V is G -equivariant. The map Φ : N → Map G, ( D, V ) ∆ ,r is continuous,because Φ is, so now Theorem 3.5 is clear.4. The space of metrics M et Preliminaries.
The following result is a standard consequence of the existence oflinear slices for smooth compact group actions (see e.g. [5, Chap. VI, § Lemma 4.1.
Let G be a finite group acting smoothly on a connected manifold X . (1) For each subgroup H ⊆ G the fixed point set X H = { x ∈ X | H ⊆ G x } is thedisjoint union of finitely many closed submanifolds of X (not necessarily of thesame dimension) satisfying T x ( X H ) = ( T x X ) H for every x ∈ X H . In particular,either X H = X or X H has empty interior. (2) Assume that the action of G on X is effective. Then X free = { x ∈ X | G x = { }} is open and dense in X . Sinks, sources, and (un)stable manifolds.
Let n > M be a compactconnected n -dimensional manifold. Suppose that M is endowed with a smooth andeffective action of a finite group Γ. Denote the stabilizer of any x ∈ M byΓ x = { γ ∈ Γ | γx = x } . Let M et denote the space of Riemannian metrics on M , and let M et Γ ⊂ M et be thesubset of Γ-invariant metrics.Let f : M → R be a Γ-invariant Morse function. This function will be fixed throughout the rest of thepaper. If p is a critical point of f , so that ∇ g f ( p ) = 0, the derivative D ∇ g f ( p ) is a welldefined endomorphism of T p M (one may define it using a connection on T M , but theresult will be independent of the chosen connection). The endomorphism D ∇ g f ( p ) is selfadjoint with respect to the Euclidean norm on T p M given by g , so D ∇ g f ( p ) diagonalizes.Denote the index of a critical point p of f by Ind f ( p ). Let Crit( f ) ⊂ M be the set ofcritical points of f , and for any k letCrit k ( f ) = { p ∈ Crit( f ) | Ind f ( p ) = k } . Define the set of sinks of f to be I = Crit n ( f ) and the set of sources to be O = Crit ( f ).The points in I (resp. O ) are the sinks (resp. sources) of the the gradient vector field ∇ g f for every g . Denote also by E = I ∪ O the collection of all local extremes of f .For any g ∈ M et and any real number t let Φ gt : M → M denote the flow at time t of ∇ g f . Define the stable and unstable manifolds of p ∈ Crit( f ) to be, respectively, W sg ( p ) = { q ∈ M | lim t →∞ Φ gt ( q ) = p } , W ug ( p ) = { q ∈ M | lim t →−∞ Φ gt ( q ) = p } . For any p ∈ E and any g ∈ M et Γ let L g ( p ) = { ψ ∈ Aut( T p M ) | ( D ∇ g f ( p )) ψ = ψ ( D ∇ g f ( p )) } = Z ( D ∇ g f ( p )) . Since Γ is finite and acts effectively on M , we can identify Γ p with a subgroup of L g ( p )using (1) in Lemma 4.1 above.4.3. The metrics in M et : generic eigenvalues at critical points. Let M et ⊂ M et Γ denote set of Γ-invariant metrics g satisfying the following conditions:(C1) for any p ∈ E the eigenvalues λ , . . . , λ n of the linearization D ∇ g f ( p ) satisfycondition (4) in Theorem 3.1;(C2) if p, q ∈ E , then the eigenvalues of D ∇ g f ( p ) and D ∇ g f ( q ) coincide if and only if p and q belong to the same Γ-orbit;(C3) for any p ∈ E we have D ∇ g f ( p ) ∈ Aut ∗ Γ p ( T p M ).Condition (C1), combined with Sternberg’s Theorem 3.1 and an easy adaptation of atheorem of Kopell [11, Theorem 6] from maps to vector fields, implies that if p ∈ I thenthe map(6) D ( p ) : Aut( ∇ g f | W sg ( p ) ) → L g ( p ) UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 13 sending any φ ∈ Aut( ∇ g f | W sg ( p ) ) to Dφ ( p ) ∈ L g ( p ) is an isomorphism (it is clear thatany such φ fixes p ); furthermore, there is a diffeomorphism h ( p ) : T p M → W sg ( p ) makingthe following diagram commutative:(7) L g ( p ) × T p M D ( p ) − × h ( p ) (cid:15) (cid:15) / / T p M h ( p ) (cid:15) (cid:15) Aut( ∇ g f | W sg ( p ) ) × W sg ( p ) / / W sg ( p ) , where the horizontal arrows are the maps defining the actions. Remark 4.2.
Strictly speaking, Sternberg’s theorem gives a diffeomorphism between aneigborhood of in T p M and a neighborhood of p in W sg ( p ) which commutes the flowsof D ∇ g f ( p ) and of ∇ g f , but such diffeomorphism can be extended uniquely imposingcompatibility with the flows to yield h ( p ) . Similarly, for any source p ∈ O the analogous map Aut( ∇ g pf | W ug ( p ) ) → L g ( p ) is anisomorphism and there is a diffeomorphism T p M → W ug ( p ) which is equivariant in theobvious sense, analogous to the case of sinks.Condition (C2) implies that for any φ ∈ Aut( ∇ g f ) and any p ∈ E we have φ ( p ) = γp for some γ ∈ Γ. Of course a priori γ may depend on p , but in the course of provingTheorem 1.1 we will deduce that for g belonging to a residual subset of M et and any φ ∈ Aut( ∇ g f ), there exists some γ such that φ ( p ) = γp for each p ∈ E .By Lemma 3.4 M et is open and dense in M et Γ . Moreover, combining (C3) withLemma 3.4 and Theorem 3.5 (together with the obvious analogue of Remark 4.2) wededuce the following result. Lemma 4.3.
Any g ∈ M et has a neighborhood U ⊂ M et such that for any p ∈ E thefollowing holds. Let V p = T p M . Endow the space of maps Map( V p , M ) with the weak(compact-open) C ∞ -topology [9, Chap 2, § . For any g ′ ∈ U there is a linear vector field X g ′ ( p ) : V p → V p depending continuously on g ′ and a Γ p -equivariant embedding h g ′ ( p ) : V p → M depending also continuously on g ′ with the following properties. (1) Z ( X g ′ ( p )) = Z ( X g ( p )) = L g ( p ) for every g ′ ∈ U , (2) h g ′ ( p ) identifies X g ′ ( p ) with the restriction of ∇ g ′ f to h g ′ ( p )( V p ) ; hence, h g ′ ( p )( V p ) = W sg ( p ) if p ∈ I and h g ′ ( p )( V p ) = W ug ( p ) if p ∈ O . Note that we do not claim that the derivative of h g ′ ( p ) at p is the identity: in fact ingeneral this will not be the case (otherwise we could not pretend to have the identifica-tions Z ( X g ′ ( p )) = Z ( X g ( p ))). The spheres S g ( p ) , the distributions A g ( p ) , and the sets F g ( p )Recall that we assume dim M >
The spheres S g ( p ) . For any g ∈ M et and any sink p ∈ I we denote by ∼ theequivalence relation in W sg ( p ) that identifies two points whenever they belong to thesame integral curve of ∇ g f . We then define S g ( p ) = ( W sg ( p ) \ { p } ) / ∼ . Let ǫ > ⊂ M be the g -geodesic sphere of radius ǫ > p . If ǫ is small enough (which we assume), then Σ is a submanifold of M diffeo-morphic to S n − and every equivalence class in S g ( p ) contains a unique representative inΣ. Hence, composing the inclusion Σ ֒ → W sg ( p ) \ { p } with the projection π p : W sg ( p ) \ { p } → S g ( p )gives a bijection Σ ≃ S g ( q ). This allows us to transport the smooth structure on Σ to asmooth structure (in particular, a topology) on S g ( p ), independent of ǫ .If g ∈ M et then the action of L g ( p ) on S g ( p ) defined via the identification (6) issmooth, and so is the natural action of Γ p on S g ( p ) (recall that M et ⊂ M et Γ ).For any p ∈ I and q ∈ O letΩ g ( p, q ) := π p ( W sg ( p ) ∩ W ug ( q )) ⊂ S g ( p ) . Since W ug ( q ) is open in M , Ω g ( p, q ) is open in S g ( p ).Similarly, if q is a source we define S g ( q ) = ( W ug ( q ) \ { q } ) / ∼ and we denote by π q : W ug ( q ) \ { q } → S g ( q )the projection. If p is a sink, then we defineΩ g ( q, p ) = π q ( W sg ( p ) ∩ W ug ( q )) , which is an open subset of S g ( q ).For convenience, if p, q ∈ I or p, q ∈ O we define Ω g ( p, q ) = ∅ .Since the fibers of the restrictions of π p and π q on W sg ( p ) ∩ W ug ( q ) are the same, thereare natural bijections σ p,qg : Ω g ( p, q ) → Ω g ( q, p ) , σ q,pg = ( σ p,qg ) − , which are easily seen to be diffeomorphisms.5.2. The singular distributions A g ( q ) . Assume through the remainder of this sectionthat g ∈ M et . We will consider, for every p ∈ E , the diagonal action of L g ( p ) on S g ( p ) k for some natural number k . If z = ( z , . . . , z k ) ∈ S g ( p ) k and ψ ∈ L g ( p ) we denote ψz = ( ψz , . . . , ψz k ) . Similarly, we will consider the diagonal extension of the maps σ p,qg : σ p,qg : Ω g ( p, q ) k → Ω g ( q, p ) k , σ p,qg z = ( σ p,qg z , . . . , σ p,qg z k ) . UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 15
We are going to use below without explicit notice analogous diagonal extensions of mapsto Cartesian products.Let n = dim M . Define(8) r := (cid:20) n n − (cid:21) . The choice of this number will be justified in the proof of Lemma 6.3.For any q ∈ E we denote by A g ( q ) ⊂ T ( S g ( q ) r ) the subspace consisting of all tangentvectors given by the infinitesimal action of the Lie algebra of L g ( q ). This gives, for any z ∈ S g ( q ) r , a linear subspace A g ( q )( z ) ⊂ T x S g ( q ) r whose dimension may vary with z (hence, one can think of A g ( q ) as a singular distribution). In concrete terms, A g ( q )( z ) = { Y g,σ ( z ) | σ ∈ Lie L g ( q ) } , where for any σ ∈ Lie L g ( q ) we denote by Y g,σ the vector field on S g ( q ) r given by theinfinitesimal action of σ .5.3. The subset F g ( q ) ⊂ S g ( q ) r . We next want to identify a dense open subset of S g ( q ) r on which the action of L g ( q ) has the smallest possible isotropy subgroup, and on which A g ( q ) restricts to a vector subbundle of T ( S g ( q ) r ). We remark that, since L g ( q ) is aninfinite group, in this situation we can not use (2) in Lemma 4.1. Let X g ( q ) := D ∇ g f ( p ) ∈ Lie L g ( q ) . Note that e t X g ( q ) corresponds, via the isomorphism D ( q ) in (6), to the flow Φ gt , so e t X g ( p ) acts trivially on S g ( q ) and hence on S g ( q ) r .Let us denote V = T q M . Then X := X g ( q ) is a diagonalizable endomorphism of V . Denote its eigenvalues by λ , . . . , λ k . Let V j ⊆ V be the subspace consisting ofeigenvectors with eigenvalue λ j . We have a decomposition V = V ⊕ · · · ⊕ V k withrespect to which we may define projections π j : V → V j . Let us say that a collection ofvectors w , . . . , w s ∈ V j is thick if s > d j = dim V j and for any 1 ≤ i < i < · · · < i d j ≤ s the vectors w i , . . . , w i dj are linearly independent. Finally, we say that a collection ofvectors v , . . . , v s ∈ V is thick if for any j the projections π j ( v ) , . . . , π j ( v s ) form a thickcollection of vectors in V j . Let G = L g ( q ). Lemma 5.1.
Suppose that v , . . . , v s is a thick collection of vectors, and that for some g ∈ G there exist real numbers t , . . . , t s satisfying gv j = e t j X v j for every j . Then g = e tX for some real number t .Proof. Consider first the case k = 1, so that X is a homothecy. Write v n +1 = a v + · · · + a n v n . The thickness condition implies that a i = 0 for every i . By assumption wehave gv i = λ i v i for some real numbers λ , . . . , λ s . In particular, λ n +1 ( a v + · · · + a n v n ) = λ a v + · · · + λ n a n v n . Taking into account that v , . . . , v n is a basis and equating coefficients we deduce that λ n +1 = λ = · · · = λ n . So the case k = 1 is proved. The case k > V j , using the fact that every g ∈ G preserves V j . (cid:3) Let S ( V ) denote the set of orbits of H = { e tX | t ∈ R } acting on V \ { } . H is acentral subgroup of G , and the action of G on V induces an action of G/H on S ( V ). Let F ⊂ S ( V ) r denote the set of tuples ( x , . . . , x r ) such that, writing x i = Hx ′ i with x ′ i ∈ V for each i , the vectors x ′ , . . . , x ′ r form a thick collection (this is independent ofthe choice of representatives x ′ i ). Lemma 5.2. (1) F is a dense an open subset of S ( V ) r ; (2) the restricted action of G/H on F is free.Proof. For (1) note that r > n , so the set F ′ of thick r -tuples in V r can be identifiedwith the complementary of finitely many proper subvarieties (those corresponding to thepossible linear relations among projections to each summand V j of subsets of the tuple,given by the vanishing of suitable determinants). Hence F ′ is a dense open subset of V r ,which implies that F ⊂ S ( V ) r is open and dense. (2) follows from Lemma 5.1. (cid:3) Assume that q is a sink. Choose a diffeomorphism h : V → W sg ( q ) making commutativethe diagram (7) with p replaced by q . Then h induces a diffeomorphism S ( V ) → S g ( q ),which can be extended linearly to S ( V ) r → S g ( q ) r . Let F g ( q ) ⊂ S g ( q ) r be the image of F under the previous diffeomorphism. The set F g ( q ) is independentof the choice of h . Indeed, two different choices of h differ by precomposition with anelement of G , and the action of G on S ( V ) r preserves F . If instead q is a source, considerthe same definition with W sg ( q ) replaced by W ug ( q ).Lemma 5.2 and an obvious estimate imply: Lemma 5.3. (1) If z ∈ F g ( q ) and ψ ∈ L g ( q ) satisfies ψz = z then ψ = e t X g ( q ) forsome t ∈ R . (2) The restriction of A g ( q ) to F is a vector bundle of rank dim G − ≤ n − . The space of metrics M et ,K We recall again that dim
M >
Definition of M et ,K . Let g ∈ M et . Let p ∈ E and let K be a natural number.Denote by k · k g the operator norm in End T p M induced by g . Denote by L g,K ( p ) ⊂ L g ( p )the subset consisting of those ψ ∈ L g ( p ) such that k ψ k g ≤ K , k ψ − k g ≤ K , and k ψ − e t X g ( p ) γ k g ≥ K − for every t ∈ R and γ ∈ Γ p . Clearly L g,K ( p ) is compact. Recall that the number r has been defined in (8) in Subsec-tion (5.2) above. For any ψ ∈ L g ( p ) we denote by α ψ : S g ( q ) r → S g ( q ) r the map given by the action of ψ . Definition 6.1.
Let p ∈ E . Define M et ,K ( p ) as the set of all metrics g ∈ M et suchthat for any ψ ∈ L g,K ( p ) there exist: (1) q, q ′ ∈ E and z ∈ Ω g ( p, q ) r satisfying ψz ∈ Ω g ( p, q ′ ) r , σ p,qg z ∈ F g ( q ) , σ p,q ′ g ψz ∈ F g ( q ′ ) , UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 17 (2) and a vector u ∈ A g ( q )( σ p,qg z ) ⊂ T σ p,qg z S g ( q ) r such that (9) D ( σ p,q ′ g ◦ α ψ ◦ σ q,pg )( u ) / ∈ A g ( q ′ )( σ p,q ′ g ◦ α ψ ( z )) . Here D ( σ p,q ′ g ◦ α ψ ◦ σ q,pg ) is the map between tangent spaces given by the differential of σ p,q ′ g ◦ α ψ ◦ σ q,pg : σ p,qg ( α − ψ (Ω g ( p, q ′ ) r )) → Ω g ( q ′ , p ) r , and σ p,qg ( α − ψ (Ω g ( p, q ′ ) r )) is an open subset of Ω g ( q, p ) containing σ p,qg z .Define finally: M et ,K = \ p ∈ E M et ,K ( p ) . M et ,K is open and dense in M et .Lemma 6.2. M et ,K ( p ) is an open subset of M et .Proof. Let g ∈ M et ,K ( p ). If ψ ∈ L g,K ( p ) ⊂ L g ( p ) and ( q, q ′ , z, u ) satisfy (1) and (2)in Definition 6.1, then we say that ( q, q ′ , z, u ) rules out ψ . The set of elements in L g ( p )which are ruled out by any given tuple ( q, q ′ , z, u ) is open. Since L g,K ( p ) is compact,it follows that there exist finitely many tuples ( q , q ′ , z , u ) , . . . , ( q ν , q ′ ν , z ν , u ν ) and opensubsets V , . . . , V ν ⊂ L g ( p ) such that L g,K ( p ) ⊂ V ∪ · · · ∪ V ν and such that, for every j ,( q j , q ′ j , z j , u j ) rules out each element of V j . Choose subsets V ′ j ⊂ V j with the propertythat L g,K ( p ) ⊂ V ′ ∪ · · · ∪ V ′ ν , and such that V ′ j is compact and contained in V j for each j .Applying Lemma 4.3 to g we deduce the existence of a neighborhood U ⊂ M et of g and natural smooth identifications S g ( q ) ≃ S g ′ ( q ) for every g ′ ∈ U and q ∈ E . Since inthe remainder of the proof we only consider metrics from U , we denote S ( q ) instead of S g ′ ( q ). We also get for every g ′ ∈ U natural isomorphisms of groups L g ( q ) ≃ L g ′ ( q ) whichare compatible with both inclusions of Γ z in L g ( q ) and L g ′ ( q ) and with the identifications S g ( q ) ≃ S g ′ ( q ), and for this reason we write L ( q ) instead of L g ′ ( q ). Now we may view V , . . . , V ν as subsets of L ( p ). Shrinking U if necessary we may assume that L g ′ ,K ( p ) ⊂ V ′ ∪ · · · ∪ V ′ ν for every g ′ ∈ U .The sets F g ′ ( q ) ⊂ S ( q ) r are independent of g ′ and the distributions A g ′ ( q ) vary contin-uously with g ′ . Similarly the subsets Ω g ′ ( p, q ) ⊂ S ( p ) vary continuously with g ′ , meaningthat, for any z ∈ Ω g ( p, q ), if g ′ is sufficiently close to g then z ∈ Ω g ′ ( p, q ) as well. Themaps σ p,qg ′ : Ω g ′ ( p, q ) → Ω g ′ ( q, p ) also depend continuously on g ′ in the obvious sense.For any g ′ ∈ U , any q ∈ E , and any z ∈ F ( q ) we define P g ′ : T z S ( q ) r → A g ′ ( q )( z ) to bethe orthogonal projection with respect to g ′ (recall that A g ′ ( q )( z ) is a vector subspaceof T z S ( q ) r ). The previous observations imply the following: for every 1 ≤ j ≤ ν thereexists a neighborhood of g , U j ⊂ U , such that for every g ′ ∈ U j and any ψ ′ ∈ V ′ j , thetuple ( q j , q ′ j , z j , P g ′ ( u j )) rules out ψ ′ . It then follows that U ∩ · · · ∩ U ν ⊂ M et ,K ( p ), andhence M et ,K ( p ) is open. (cid:3) Lemma 6.3. M et ,K ( p ) is a dense subset of M et .Proof. We will use the following lemma, whose proof is postponed to the Appendix.
Lemma 6.4.
Suppose that g ∈ M et Γ , x ∈ M \ Crit( f ) and y = Φ gt ( x ) for some nonzero t . Suppose that the stabilizer Γ x is trivial. Let v ∈ T x M be a nonzero vector, and let w = D Φ gt ( x )( v ) . Given any u ∈ T v ( T M ) and any Γ -invariant open subset U ⊂ M containing Φ gt ′ ( x ) for some t ′ ∈ (0 , t ) , there exists some g ′ ∈ C ∞ ( M, S T ∗ M ) Γ supportedon U such that ∂∂ǫ D Φ g + ǫg ′ − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = u. Fix some g ∈ M et . We assume for concreteness throughout the proof that p is a sink.The case in which p is a source follows from the same arguments (or replacing f by − f ).We claim that the set of points in S g ( p ) with trivial stabilizer in Γ is open and dense.Indeed, on the one hand the points in W sg ( p ) with trivial stabilizer form an open anddense subset, thanks to (2) in Lemma 4.1 and the openness of W sg ( p ) ⊂ M and on theother hand the stabilizer of any z ∈ W sg ( p ) is equal to the stabilizer of the point in S g ( p )it represents, because the action of Γ preserves f and the restriction of f to the fibers ofthe projection W sg ( p ) \ { p } → S g ( p ) is injective.Let ψ ∈ L g,K ( p ). Since ψ does not belong to Γ p ⊂ L g ( p ) and since the union of the sets { Ω( p, q ) } q ∈ O is an open and dense subset of S g ( p ), there exist q, q ′ ∈ O (not necessarilydistinct) and a point c ∈ Ω( p, q ) satisfying ψc ∈ Ω( p, q ′ ) and ψc = γc for every γ ∈ Γ p .By the previous claim, we can also assume that the stabilizer of c is trivial.Choose a metric on the sphere S g ( p ). For any y ∈ S g ( p ) and any r > B S,r ( y ) the closed ball in S g ( p ) of radius r and center y . Choose ǫ > γB S,ǫ ( c ) ∩ ψB S,ǫ ( c ) = ∅ for every γ ∈ Γ p , and in such a way that B S,ǫ ( c ) ⊂ Ω g ( p, q ) and ψB S,ǫ ( c ) ⊂ Ω g ( p, q ′ ). Replacing ǫ by a smaller number if necessary we canassume that the stabilizers of the points in B S,ǫ ( c ) are all trivial. Take r distinct points z ψ , . . . , z ψr ∈ B S,ǫ/ ( c ) and tangent vectors u ψi ∈ T σ p,qg ( z ψi ) S g ( q ) for i = 1 , . . . , r . Letting z ψ = ( z ψ , . . . , z ψr ), we may assume that σ p,qg ( z ψ ) ∈ F g ( q ) and σ p,q ′ g ( ψz ψ ) ∈ F g ( q ′ ) because F g ( q ) (resp. F g ( q ′ )) is dense in S rg ( q ) (resp. S rg ( q ′ )).Let O ψ be the set of all elements ψ ′ ∈ L g ( p ) satisfying γB S,ǫ/ ( c ) ∩ ψ ′ B S,ǫ/ ( c ) = ∅ forevery γ ∈ Γ p and σ p,q ′ g ( ψ ′ z ψ ) ∈ F g ( q ′ ). Clearly O ψ ⊂ L g ( p ) is open and contains ψ .Denote the open ball in M with center x and radius δ by B δ ( x ). Take real numbers a < b < f ( p ) in such a way that [ a, f ( p )) does not contain any critical value of f . Take δ > B δ ( p ) (resp. B δ ( q ), B δ ( q ′ )) is entirely contained in W sg ( p )(resp. W ug ( q ) and W ug ( q ′ )) and inf f | B δ ( p ) > b (resp. sup f | B δ ( q ) < a and sup f | B δ ( q ′ ) < a ).Pick, for each 1 ≤ i ≤ r , points x i ∈ B δ ( p ) \ { p } and y i ∈ B δ ( q ) \ { q } both representing z ψi ∈ S g ( p ), and a tangent vector v i ∈ T x i M projecting to u ψi ∈ T z i S g ( p ). Define realnumbers t , . . . , t r by the condition that y i = Φ gt i ( x i ), and let w i = D Φ gt i ( v i ).Let U ⊂ M be an open Γ-invariant subset contained in f − (( a, b )) ∩ W sg ( p ) whoseprojection to S g ( p ) contains z ψ , . . . , z ψr and is disjoint from ψ ( B S,ǫ/ ( c )). By Lemma6.4 one can pick a finite dimensional vector subspace G ψ ⊂ C ∞ ( M, S T ∗ M ) Γ , all of whose elements are supported in U , with the property that the linear map(10) G ψ ∋ g ′ (cid:18) ∂∂ǫ D Φ g + ǫg ′ − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 , . . . , ∂∂ǫ D Φ g + ǫg ′ − t r ( w r ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 (cid:19) ∈ r M i =1 T v i ( T M ) UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 19 is surjective.Choose an open neighborhood O ′ ψ ⊂ O ψ of ψ whose closure in O ψ is compact. Since L g,K ( p ) is compact there exist ψ , . . . , ψ s ∈ L g,K ( p ) such that L g,K ( p ) ⊂ O ′ ψ ∪ · · · ∪ O ′ ψ s .Denote z i = z ψ i ∈ S g ( p ) r and u i = u ψ i ∈ T ( S g ( p ) r ).Let G = P i G ψ i . Let M be the set of all g ′ ∈ M et Γ satisfying the following conditions:(1) g ′ − g ∈ G ,(2) σ p,qg ′ ( z i ) ∈ F g ′ ( q ) = F g ( q ) for every i ,(3) σ p,q ′ g ′ ( ψz i ) ∈ F g ′ ( q ′ ) = F g ( q ′ ) for every i and every ψ ∈ O ′ ψ i .To explain conditions (2) and (3), note that since g ′ − g ∈ P i G ψ i and the elementsin each G ψ i are supported away from the critical points, we can canonically identify S g ( q ) = S g ′ ( q ) and S g ( q ′ ) = S g ′ ( q ′ ), and similarly F g ( q ) = F g ′ ( q ) and F g ( q ′ ) = F g ′ ( q ′ ).Note that { g ′ − g | g ′ ∈ M } can be identified with an open subset of G containing 0,so M has a natural structure of (finite dimensional) smooth manifold.Consider, for each i ∈ { , . . . , s } , V i = { ( g ′ , ψ ′ , b ) ∈ M × O ψ i × T ( S g ( q ′ ) r ) | b = D ( σ p,q ′ g ′ ◦ α ψ ′ ◦ σ q,pg ′ )( u i ) } and its subvariety V ′ i = { ( g ′ , ψ ′ , b ) ∈ M × O ′ ψ i × T ( S g ( q ′ ) r ) | b = D ( σ p,q ′ g ′ ◦ α ψ ′ ◦ σ q,pg ′ )( u i ) } . Let also A = M × L g ( p ) × A g ( q ′ ) | F g ( q ′ ) . Note that V i , V ′ i , A are subvarieties of M × L g ( p ) × T ( S g ( q ′ ) r ).Let N i = V i ∩ A ∩ ( { g } × O ψ i × T ( S g ( p ) r ). The definition of G ψ i guarantees that V i and A intersect transversely along N i . Consequently, there exists a neighborhood of N i , N i ⊂ M × O ψ i × T ( S g ( q ′ ) r ) , such that the intersection V i ∩ A ∩ N i is a smooth manifold whose dimension satisfies d − dim( V i ∩ A ∩ N i ) = min { d + 1 , ( d − dim V i ) + ( d − dim A ) } , where d = dim M × L p ( g ) × T ( S g ( q ′ ) r ) . This formula is consistent with the convention that a set is empty if and only if itsdimension is −
1. Consider the projection π i : V i ∩ A → M . Since the closure of V ′ i inside V i is compact, there exists a neighborhood of g , M i ⊂ M ,with the property that π − i ( M i ) ∩ V ′ i ⊂ N i . Hence, π − i ( M i ) ∩ V ′ i is a smooth manifold.Let M reg i ⊂ M i be the set of regular values of π i restricted to π − i ( M i ). We claim that for every g ′ ∈ M reg i we have π − i ( g ′ ) ∩ V ′ i = ∅ . To prove the claim itsuffices to check that dim π − i ( g ′ ) ∩ V ′ i = −
1. Now,dim π − i ( g ′ ) ∩ V ′ i = dim( V i ∩ A ∩ N i ) − dim M = d − min { d + 1 , ( d − dim V i ) + ( d − dim A ) } − dim M . If ( d − dim V i ) + ( d − dim A ) ≥ d + 1 then this is clearly negative. So assume thatinstead ( d − dim V i ) + ( d − dim A ) < d + 1. Since the projection of V i to M × O ψ i is adiffeomorphism, we have d − dim V i = d − (dim M × O ψ i ) = d − (dim M × L g ( p )) = dim T ( S g ( q ′ ) r ) = 2 r ( n − . On the other hand we have, using (2) in Lemma 5.3, d − dim A = dim T ( S g ( q ′ ) r ) − dim A g ( q ′ ) | F g ( q ′ ) ≥ r ( n − − ( r ( n −
1) + n −
1) = r ( n − − n + 1 . Combining both estimates we compute:dim π − i ( g ′ ) ∩ V ′ i ≤ d − r ( n − − r ( n −
1) + n − − dim M = dim L p ( g ) × T ( S g ( q ′ ) r ) − r ( n −
1) + n − ≤ n + 2 r ( n − − r ( n −
1) + n −
1= 2 n − r ( n − − . Our choice of r , see (8), implies that 2 n − r ( n − − <
0, so the claim is proved.Finally, let M reg = M reg1 ∩ · · · ∩ M reg s . We claim that M reg ⊂ M et ,K ( p ). Indeed, suppose that g ′ ∈ M reg and let ψ ∈ L g,K ( p )be any element. Then ψ ∈ O ′ ψ i for some i and we have, on the one hand, z i ∈ S g ′ ( p, q ) r , ψz i ∈ S g ′ ( p, q ′ ) r , σ p,qg ′ ( z i ) ∈ F g ′ ( q ) , σ p,q ′ g ′ ( ψz i ) ∈ F g ′ ( q ′ ) , and, on the other hand, the fact that π − i ( g ′ ) ∩ V ′ i = ∅ implies that D ( σ p,q ′ g ′ ◦ α ψ ◦ σ q,pg ′ )( u i ) / ∈ A g ′ ( q ′ )( σ p,q ′ g ′ ◦ α ψ ′ ( z i )) . This proves the claim.Sard’s theorem (see e.g. [9, Chap 3, § M reg is residual in M . Hence M reg is dense in a neighborhood of g ∈ M , so M et ,K is dense in a neighborhood of g . (cid:3) Recall that M et ,K = T p ∈ E M et ,K ( p ). The preceding two lemmas imply: Lemma 6.5. M et ,K is a dense an open subset of M et . Proof of Theorem 1.1 for dim
M > M et f = M et ∩ \ K ∈ N M et ,K . Since each of the sets appearing in the right hand side of the equality is open and densein M et Γ (see Subsection 4.3 and Lemma 6.5), M et f is a residual subset of M et Γ . Fix UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 21 some g ∈ M et f and let φ ∈ Aut( ∇ g f ). We are going to check that there exists some γ ∈ Γ and some t ∈ R such that φ ( x ) = Φ gt ( γ x )for every x ∈ M . This will prove Theorem 1.1. Lemma 7.1.
For each p ∈ I (resp. p ∈ O ) there exists some γ ∈ Γ and some t ∈ R suchthat φ ( x ) = Φ gt ( γ x ) for every x ∈ W sg ( p ) (resp. for every x ∈ W ug ( p ) ).Proof. Suppose that p ∈ I (the case p ∈ O is dealt with in the same way with the obviousmodificatoins). By property (C2) in the definition of M et (see Subsection 4.3) thereexists some γ ∈ Γ such that φ ( p ) = γ p . Hence, up to composing φ with the action of γ ,we can (and do) suppose that φ ( p ) = p .Once we know that φ fixes p , we conclude that it restricts to a diffeomorphism of W sg ( p )preserving ∇ g f , which we identify with an element φ p ∈ L g ( p ) via the isomorphism (6).Next, let us prove that the action of φ p on S g ( p ) coincides with the action of some γ ∈ Γ p .If this is not the case, then φ p ∈ L g,K ( p ) for some natural K (see Subsection 6.1). Since g ∈ M et ,K , it follows that there exist sources q, q ′ ∈ O and z ∈ Ω( p, q ) r satisfying φ p z ∈ Ω g ( p, q ′ ) r , σ p,qg z ∈ F g ( q ) , σ p,q ′ g φ p z ∈ F g ( q ′ ) , and a vector u ∈ A g ( q )( σ p,qg ( z )) satisfying(11) D ( σ p,q ′ g ◦ α φ p ◦ σ q,pg )( u ) / ∈ A g ( q ′ )( σ p,q ′ g ◦ α φ p ( z )) . By the definition of A g ( q ), we may write u = Y g,s ( σ p,qg z ) for some s ∈ Lie L g ( q ).The fact that z ∈ Ω( p, q ) r and φ p z ∈ Ω g ( p, q ′ ) r implies that φ ( q ) = q ′ , so φ maps W ug ( q ) diffeomorphically to W ug ( q ′ ); since φ preserves ∇ g f , φ induces by conjugation anisomorphism ψ : L g ( q ) → L g ( q ′ ) . The corresponding map at the level of Lie algebras associates to s an element ψ ( s ) ∈ Lie L g ( q ′ ), and in fact we have D ( σ p,q ′ g ◦ α φ p ◦ σ q,pg )( u ) = D ( σ p,q ′ g ◦ α φ p ◦ σ q,pg )( Y g,s ( σ p,qg z ))= Y g,ψ ( s ) ( σ p,q ′ g ( φ p z )) . (12)The last expression manifestly belongs to A g ( q ′ )( σ p,q ′ g ◦ α φ p ( z )), and this contradicts (11).So we have proved that there is some γ ∈ Γ p such that γ − φ p acts trivially on S g ( p ).Now statement (1) in Lemma 5.3 implies that γ − φ p = e t X g ( p ) for some t ∈ R , so we maywrite φ p = γe t X g ( p ) or, equivalently, that φ p ( y ) = Φ gt ( γ y ) for every y ∈ W sg ( p ). (cid:3) For any p ∈ E we denote W g ( p ) := W sg ( p ) (resp. W g ( p ) := W ug ( p )) if p ∈ I (resp. if p ∈ O ). Now the proof of the case dim M > { t p ∈ R } p ∈ E and { γ p ∈ Γ } p ∈ E such that φ ( x ) = Φ gt p ( γ p x ) for every p ∈ E and x ∈ W g ( p ). The firststep consists in proving that if all γ p ’s are equal then all t p ’s are equal as well (this is [22,Lemma 5]). The second step consists on reducing the general case to the one covered bythe first step. This is explained in the three paragraphs following [22, Lemma 5]. Appendix A. Change of the gradient flow as the metric varies
Recall what we want to prove.
Lemma A.1.
Suppose that g ∈ M et Γ , x ∈ M \ Crit( f ) and y = Φ gt ( x ) for some nonzero t . Suppose that the stabilizer Γ x is trivial. Let v ∈ T x M be a nonzero vector, and let w = D Φ gt ( x )( v ) . Given any u ∈ T v ( T M ) and any Γ -invariant open subset U ⊂ M containing Φ gt ′ ( x ) for some t ′ ∈ (0 , t ) , there exists some g ′ ∈ C ∞ ( M, S T ∗ M ) Γ supportedon U such that (13) ∂∂ǫ D Φ g + ǫg ′ − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = u. We will prove Lemma A.2 using the following weaker version of it.
Lemma A.2.
Let g, x, t, y be as in Lemma A.1. Given any v ∈ T x M and any Γ -invariant open subset U ⊂ M containing Φ gt ′ ( x ) for some t ′ ∈ (0 , t ) , there exists some g ′ ∈ C ∞ ( M, S T ∗ M ) Γ supported on U such that ∂∂ǫ Φ g + ǫg ′ − t ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = v. Before proving Lemma A.2 we prove two auxiliary lemmas.
Lemma A.3.
Let g ∈ M et Γ . Let Y ∈ C ∞ ( M ; T M ) Γ satisfy supp Y ∩ Crit( f ) = ∅ . Thereexists some δ > and a smooth map G : ( − δ, δ ) → M et Γ such that G (0) = g and, forevery ǫ ∈ ( − δ, δ ) , ∇ G ( ǫ ) f = ∇ g f + ǫY. Proof.
This is a consequence of the following elementary fact in linear algebra. Let V be a finite dimensional real vector space and let α ∈ V ∗ be a nonzero element. Let E ⊂ S V ∗ be the open subset of Euclidean pairings, and let ∇ : E → V be the mapdefined by the property that e ( ∇ ( e ) , u ) = α ( u ) for every u ∈ V . Then ∇ is a submersionwith contractible fibers. (cid:3) For any vector field X on M we denote by Φ Xt : M → M the flow at time t of X . Lemma A.4.
Let
X, Y ∈ C ∞ ( M ; T M ) . Suppose that p / ∈ supp Y and that X has noregular periodic integral curve. For any t we have ∂∂s Φ X + sL X Yt ( p ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Y (Φ Xt ( p )) . Proof. If X ( p ) = 0 then the formula is immediate. So suppose that X ( p ) = 0. Then Z := { Φ Xτ | ≤ τ ≤ t } is diffeomorphic to [0 , X has no regular periodic integralcurve. Take an open neighborhood U ⊂ M of Z and coordinates x = ( x , . . . , x n ) : U → R n with respect to which p = (0 , . . . ,
0) and X = ∂∂x , so that Φ Xτ ( x , . . . , x n ) =( x + τ, x , . . . , x n ). Suppose that x ∗ ( L X Y | U ) = P a j ∂∂x j and that x ∗ ( Y | U ) = P b j ∂∂x j .Since Y ( p ) = 0, we have b j ( t, , . . . ,
0) = R t a j ( τ, , . . . , dτ for every j . Let γ ( t, s ) = UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 23 x (Φ X + sL X Yt ( p )). Let e , . . . , e n denote the canonical basis of R n . We have ∂∂t ∂γ ( t, s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∂∂s ∂γ ( t, s ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∂∂s (cid:16) (1 , , . . . ,
0) + s X a j ( γ ( t, s )) e j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) s =0 = X a j ( γ ( t, e j = X a j ( t, , . . . , e j . Consequently, ∂γ ( t, s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 = X (cid:18)Z t a j ( τ, , . . . , dτ (cid:19) e j = X b j ( t, , . . . , e j = x ∗ ( Y (Φ Xt ( p ))) , which proves the desired formula. (cid:3) Let us now prove Lemma A.2. Fix some g ∈ M et Γ , let x ∈ M \ Crit( f ), and let y = Φ gt ( x ) for some nonzero t . Suppose that the stabilizer Γ x is trivial, let v ∈ T x M be any element and let U ⊂ M be a Γ-invariant open subset containing Φ gt ′ ( x ) for some t ′ ∈ (0 , t ). Let X = ∇ g f . Since x / ∈ Crit( f ) and Γ x is trivial, there exists an invariantvector field Y ∈ C ∞ ( M ; T M ) Γ whose support is contained in U \ Crit( f ) and whichsatisfies Y ( x ) = v . By Lemma A.4 we have ∂∂s Φ X + sL X Y − t ( y ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Y ( x ) . By Lemma A.3 there exists some δ > { g ǫ } ǫ ∈ ( − δ,δ ) satisfying g = g and ∇ g ǫ f = ∇ g f + ǫY for every ǫ ∈ ( − δ, δ ). Setting g ′ = ∂g ǫ /∂ǫ | ǫ =0 it followsthat ∂∂ǫ Φ g + ǫg ′ − t ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = v. Finally we prove Lemma 6.4/A.1.Let π : T M → M denote the projection and let Dπ : T ( T M ) → T M denote itsderivative. A vector field X on M defines a vector field e X on T M by the condition that(14) D Φ Xt = Φ e Xt for every t . In particular, Φ e X − t ( w ) = v .We will use these properties of the map X e X : (1) it is linear, (2) Dπ ( e X ( u )) = X ( π ( u )) for every vector field X and any u ∈ T M , (3) if X ( a ) = 0 for some point a ∈ M ,then the restriction of e X to T a M is vertical and can be identified with the linear vectorfield T a M → T a M given by the endomorphism DX ( a ) of T a M , and (4) it is compatiblewith Lie brackets: ^ [ X, Y ] = [ e X, e Y ].Let g ∈ M et Γ , let x ∈ M \ Crit( f ) be a point with trivial stabilizer, and let y = Φ gt ( x )for some nonzero t . Let v ∈ T x M be nonzero and let w = D Φ gt ( x )( v ). Finally, supposegiven u ∈ T v ( T M ) and a Γ-invariant open subset U ⊂ M containing Φ gt ′ ( x ) for some t ′ ∈ (0 , t ).Let X = ∇ g f . Since x / ∈ Crit( f ) and Γ x is trivial, there exists an invariant vectorfield Y ∈ C ∞ ( M ; T M ) Γ whose support is contained in U \ Crit( f ) and which satisfies Y ( x ) = Dπ ( u ). Then u = e Y ( v ) − u satisfies Dπ ( u ) = 0. Let L : T x M → T x M bea linear map satisfying Lv = u , and let Y ∈ C ∞ ( M ; T M ) Γ have support contained in U \ Crit( f ) and satisfy Y ( x ) = 0 and DY ( x ) = L . Let Y = Y + Y . Then e Y ( v ) = u . By (14), the properties of the map X e X , and Lemma A.4 (with M replaced by T M in the statement), we have ∂∂s D Φ X + sL X Y − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∂∂s Φ ^ X + sL X Y − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∂∂s Φ e X + sL e X e Y − t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = e Y ( v ) . By Lemma A.3 there exists some δ > G : ( − δ, δ ) → M et Γ satisfying G (0) = g and ∇ G ( ǫ ) f = ∇ g f + ǫY for every ǫ ∈ ( − δ, δ ). Then g ′ = G ′ (0) satisfies (13). Appendix B. Glossary
Here we list in alphabetical order some of the symbols used in Section 3 and the nextones. A g ( p ) = the distribution on S g ( p ) r given by the infinitesimal diagonal actionof the Lie algebra of L g ( p ), see Subsection 5.2 α ψ = the map S g ( p ) r → S g ( p ) r given by the action of ψ ∈ L g ( p ) E = I ∪ O f : M → R = a Γ-invariant Morse function on M , see Subsection 4.2 F g ( p ) = a dense open subset of S g ( p ) r on which the only elements of L g ( p ) withfixed points are those of the form D Φ tg ( p ), see Subsection 5.3Φ tg = the time t gradient flow of f w.r.t. the metric g Γ = a finite group acting smoothly and effectively on M I = the critical points of f of index n = dim M (sinks of ∇ g f ) L g ( p ) = the automorphisms of T p M commuting with D ∇ g f ( p ), see Subsection 4.2;if p ∈ I and g ∈ M et then L g ( p ) is naturally isomorphic to Aut( ∇ g f | W sg ( p ) )(if p ∈ O then the same holds for W ug ( p )), see Subsection 4.3 M = a compact connected smooth manifold of dimension at least 2 M et = the set of Γ-invariant metrics on M defined in Subsection 4.3 M et ,K = the set of Γ-invariant metrics on M defined in Section 6 O = the critical points of f of index 0 (sources of ∇ g f )Ω g ( p, q ) = the projection to S g ( q ) of W sg ( p ) ∩ W ug ( q ) if p ∈ I and q ∈ O , and theprojection of W ug ( p ) ∩ W sg ( q ) if p ∈ O and q ∈ I , see Subsection 5.1 S g ( p ) = the set of nonconstant integral curves of ∇ g f | W sg ( p ) (resp. ∇ g f | W ug ( p ) )if p ∈ I (resp. p ∈ O ), see Subsection 5.1 σ p,qg = the natural isomorphism Ω g ( p, q ) → Ω g ( q, p ), see Subsection 5.1 W sg ( p ) = the stable set of p ∈ I W ug ( q ) = the unstable set of q ∈ O UTOMORPHISMS OF GENERIC GRADIENT VECTOR FIELDS WITH SYMMETRIES 25
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