Homogeneous affine surfaces: affine Killing vector fields and Gradient Ricci solitons
aa r X i v : . [ m a t h . DG ] D ec HOMOGENEOUS AFFINE SURFACES: AFFINE KILLINGVECTOR FIELDS AND GRADIENT RICCI SOLITONS
M. BROZOS-V ´AZQUEZ E. GARC´IA-R´IO, AND P. GILKEY
Abstract.
The homogeneous affine surfaces have been classified by Opozda.They may be grouped into 3 families, which are not disjoint. The connectionswhich arise as the Levi-Civita connection of a surface with a metric of constantGauss curvature form one family; there are, however, two other families. For asurface in one of these other two families, we examine the Lie algebra of affineKilling vector fields and we give a complete classification of the homogeneousaffine gradient Ricci solitons. The rank of the Ricci tensor plays a central rolein our analysis. Introduction
Homogeneity.
The notion of homogeneity is central in geometry. In order tomake precise the level of homogeneity one usually refers to the underlying structure.In pseudo-Riemannian geometry, local homogeneity means that for any two pointsthere is a local isometry sending one point to the other. If an additional structure(K¨ahler, contact, etc.) is considered on the manifold, then one further assumes thatthis structure is preserved by the local isometries. In the affine setting, homogeneitymeans that for any two points there is an affine transformation sending one pointinto the other. There is an intermediate level of homogeneity which was exploredin [7, 12]. A pseudo-Riemannian manifold may be locally affine homogeneous butnot locally homogeneous, i.e., for any two points there exists a (not necessarilyisometric) transformation sending one point to the other which preserves the Levi-Civita connection.Homogeneous affine surfaces were studied from a local point of view by severalauthors. A complete description was first given in [11] for the special case when theRicci tensor is skew-symmetric. The general situation was later addressed in [14],where Opozda obtained the local form of the connection of any locally homogeneousaffine surface. More recently, Opozda’s result was generalized in [2] to the moregeneral case of connections with torsion. The above classification results have beenextensively used both in the affine and the pseudo-Riemannian setting, where oneuses the Riemannian extension to relate affine and pseudo-Riemannian geometry.1.2.
Notational conventions.
An affine manifold is a pair M = ( M, ∇ ) where ∇ is a torsion free connection on the tangent bundle of a smooth manifold M ofdimension m . Let ~x = ( x , . . . , x m ) be a system of local coordinates on M . Weadopt the Einstein convention and sum over repeated indices to expand: ∇ ∂ xi ∂ x j = Γ ij k ∂ x k in terms of the Christoffel symbols
Γ = Γ ∇ := (Γ ij k ); the condition that ∇ is torsionfree is then equivalent to the symmetry Γ ij k = Γ jik . The curvature operator R , the Mathematics Subject Classification.
Key words and phrases.
Homogeneous affine surface, affine Killing vector field, affine gradientRicci soliton, affine gradient Yamabe soliton, Riemannian extension.Supported by projects EM2014/009, GRC2013-045 and MTM2013-41335-P with FEDERfunds (Spain).
Ricci tensor ρ , and the symmetric Ricci tensor ρ s are given, respectively, by setting R ( ξ , ξ ) := ∇ ξ ∇ ξ − ∇ ξ ∇ ξ − ∇ [ ξ ,ξ ] ,ρ ( ξ , ξ ) := Tr { ξ → R ( ξ , ξ ) ξ } , and ρ s ( ξ , ξ ) := ( ρ ( ξ , ξ ) + ρ ( ξ , ξ )) . Locally homogeneous affine surfaces.
Let M = ( M, ∇ ) be an affine sur-face. We say that M is locally homogeneous if given any two points of M , there isthe germ of a diffeomorphism Φ taking one point to another with Φ ∗ ∇ = ∇ . Onehas the following classification result due of Opozda [14]: Theorem 1.1.
Let M = ( M, ∇ ) be a locally homogeneous affine surface. Then atleast one of the following three possibilities holds which describe the local geometry: (1) There exist local coordinates ( x , x ) so that Γ ijk = Γ jik is constant. (2) There exist local coordinates ( x , x ) so that Γ ijk = ( x ) − C ij k where C ijk = C jik is constant. (3) ∇ is the Levi-Civita connection of a metric of constant sectional curvature. Definition 1.2.
An affine surface M is said to be Type A (resp. Type B or Type C )if M is locally homogeneous, if M is not flat, and if Assertion 1 (resp. Assertion 2or Assertion 3) of Theorem 1.1 holds. Let F A := {M = ( R , ∇ ) : Γ ∇ constant and ∇ not flat } , F B := {M = ( R + × R , ∇ ) : Γ ∇ = ( x ) − C for C constant and ∇ not flat } . Let
M ∈ F A . We will show in Lemma 2.2 that ρ is symmetric. Since M is notflat, Rank { ρ } 6 = 0. We therefore may decompose F A = F A ∪ F A where F A ν := {M ∈ F A : Rank { ρ } = ν } . The affine surfaces in the family F A (resp. F B ) form natural models for the Type A (resp. Type B ) surfaces and we will often work in this context.Surfaces of Type A and Type B can have quite different geometric properties.The Ricci tensor of any Type A surface is symmetric; this can fail for a Type B surface. Thus the geometry of a Type B surface is not as rigid as that of a Type A surface; this is closely related to the existence of non-flat affine Osserman structures[6, 8]. Any Type A surface is projectively flat; this can fail for a Type B surface.The local geometry of any Type A surface can be realized on a compact torus[9, 15]; this can also fail for a Type B geometry. Remark 1.3. If M = R and if the Christoffel symbols Γ of ∇ are constant, then R acts transitively on M by translations and this group action preserves ∇ . Thusevery element of F A is affine homogeneous. If M = R + × R and if the Christoffelsymbols of ∇ have the form Γ = ( x ) − C for C constant, then the ax + b groupacts transitively on M by ( a, b ) : ( x , x ) → ( ax , ax + b ) for a > ∇ . Thus every element of F B is affine homogeneous. Thesetwo structure groups (which up to isomorphism are the only two simply connected2-dimensional Lie groups) will play an important role in our analysis. Remark 1.4.
The three possibilities of Theorem 1.1 are not exclusive as we shallsee presently. In Theorem 3.11, we will identify the local geometries which are bothType A and Type B and also the local geometries which are both type Type B andType C . There are no surfaces which are both Type A and Type C . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 3
Outline of the paper.
In Section 2, we use the action of the natural structuregroups on the families F A and F B to partially normalize the Christoffel symbols.Let M be a Type- A surface with Rank( ρ ) = 1. In Lemma 2.5, we will define α ( M )and show it is an affine invariant in this setting. Subsequently, in Theorem 3.8, wewill show that α identifies the moduli space of such surfaces with ρ ≥ , ∞ )and with ρ ≤ −∞ , B ge-ometries in Lemma 2.8. Lemma 2.10 provides a complete characterization of theelements of F B where ρ is symmetric, recurrent, and of rank 1, and where ∇ ρ issymmetric. This will play a central role in our identification of the affine surfaceswhich are both Type A and Type B .Section 3 is devoted to the study of the Lie algebra K ( M ) of affine Killing vectorfields. Let M be an affine surface. In Lemma 3.1, we will show if M is homogeneous,then 2 ≤ dim { K ( M ) } ≤
6; the extremal case where dim { K ( M ) } = 6 occurs only if M is flat. We shall exclude the flat setting from consideration henceforth.Let M ∈ F A . To simplify the notation, we set ∂ := ∂ x and ∂ := ∂ x . Let K A := Span { ∂ , ∂ } be the Lie algebra of the translation group R . By Remark 1.3, K A ⊂ K ( M ). In Theorem 3.4, we show dim { K ( M ) } > ρ has rank1 and that dim { K ( M ) } = 4 in this setting. In Theorem 3.8, we exhibit invariantswhich completely detect the local isomorphism class of a Type A affine surfacewith Rank { ρ } = 1, we also determine which Type A surfaces are also of Type B ,and we give the abstract structure of the (local) Lie algebras involved using theclassification of Patera et. al [16]; representatives of these classes are given inLemma 3.6.Let M ∈ F B . We will show that dim { K ( M ) } ∈ { , , } in Section 3.2; M is also of Type A if and only if dim { K ( M ) } = 4. This characterizes the localgeometries which are the intersection of Type A and Type B . The geometrieswhich are of both Type B and of Type C form a proper subset of those surfaceswhere dim { K ( M ) } = 3.The Hessian H ∇ f of f ∈ C ∞ ( M ) is the symmetric 2-tensor H ∇ f := ∇ ( df ) = f ; ij dx i ◦ dx j . If g is a pseudo-Riemannian metric on M , let H gf := H ∇ g f be the Hessian which isdefined by the Levi-Civita connection ∇ g and let ρ g be the associated Ricci tensor. Definition 1.5.
Let M be a smooth manifold, let ∇ be a torsion free connectionon M , let g be a pseudo-Riemannian metric on M , let τ be the scalar curvature of g , and let f ∈ C ∞ ( M ) be a smooth function on M . We say that(1) ( M, ∇ , f ) is an affine gradient Yamabe soliton if H ∇ f = 0. Let Y ( M ) bethe space of functions on M so that ( M, ∇ , f ) is an affine gradient Yamabesoliton; Y ( M ) = ker( H ∇ ).(2) ( M, ∇ , f ) is an affine gradient Ricci soliton if H ∇ f + ρ s = 0. Let A ( M )be the space of functions on M so that ( M, ∇ , f ) is an affine gradientRicci soliton. If A ( M ) is non-empty, then A ( M ) = f + Y ( M ) for any f ∈ A ( M ).(3) ( M, g, f ) is a gradient Yamabe soliton if there exists λ ∈ R so H gf = ( τ − λ ) g .(4) ( M, g, f ) is a gradient Ricci soliton if there exists λ ∈ R so H gf + ρ g = λg .If λ = 0, then the soliton is said to be steady .(5) A soliton is said to be trivial if the potential function f is constant.There is a close connection between affine geometry and neutral signature geome-try. Let M = ( M, ∇ ) be an affine manifold and let ( x , . . . , x m ) be local coordinateson M . Express ω = y i dx i to introduce the dual fiber coordinates ( y , . . . , y m ) on M. BROZOS-V´AZQUEZ E. GARC´IA-R´IO, AND P. GILKEY the cotangent bundle T ∗ M . Let φ = φ ij be a symmetric 2-tensor on M . The deformed Riemannian extension g ∇ ,φ is the metric of neutral signature ( m, m ) on T ∗ M given by g ∇ ,φ = dx i ⊗ dy i + dy i ⊗ dx i + ( φ ij − y k Γ ij k ) dx i ⊗ dx j . It is invariantly defined, i.e. it is independent of the particular coordinate systemchosen. The following result [1, 3] provided our initial motivation for examiningaffine gradient Ricci solitons in the 2-dimensional setting; we state the results forgradient Ricci solitons and Yamabe solitons in parallel to simplify the exposition:
Theorem 1.6.
Let ( N, g, F ) be a non-trivial self-dual gradient Ricci (resp. Yam-abe) soliton of neutral signature (2 , . (1) If k dF k 6 = 0 at a point P ∈ N , then ( N, g ) is locally isometric to a warpedproduct I × ψ N where N is a 3-dimensional pseudo-Riemannian manifoldof constant sectional curvature (resp. scalar curvature). (2) If k dF k = 0 on N , then ( N, g ) is locally isometric to the cotangent bundle T ∗ M of an affine surface ( M, ∇ ) equipped with the deformed Riemannianextension g ∇ ,φ . Furthermore, the potential function of the soliton is of theform F = f ◦ π , for some function f on M so that ( M, ∇ , f ) is an affinegradient Ricci (resp. Yamabe) soliton. In Section 4, we examine affine gradient Ricci solitons if M is Type A and/orType B . Let f be the potential function of an affine gradient Ricci soliton andlet X ∈ K ( M ). In Lemma 4.1, we show that X ( f ) is the potential function of anaffine gradient Yamabe soliton. Thus affine gradient Ricci solitons and Yamabesolitons are closely linked concepts. Using this fact, we analyze the existence ofaffine gradient Ricci and Yamabe solitons on homogeneous affine surfaces. Notunexpectedly, Type A and Type B affine connections behave differently.Let M ∈ F A . In Theorem 4.3, we show M is a gradient Ricci soliton if and onlyif Rank { ρ } = 1 or, equivalently in view of the results of Section 3, dim { K ( M ) } > F B which have skew-symmetric Ricci tensor or, equivalently,so that ( T ∗ M, g ∇ ,φ ) is Ricci flat and hence are trivial Ricci solitons. In Theorem 4.9and Theorem 4.10, we give elements of F B which are non-trivial affine gradientRicci solitons and which are not of Type A . Finally, Theorem 4.12 gives a completeclassification, up to affine equivalence, of homogeneous affine gradient Ricci solitons.The associated deformed Riemannian extensions then form a large family of non-conformally flat self-dual gradient Ricci and Yamabe solitons.1.5. Local versus global geometry.
There is always a question of the localversus the global geometry of an object in differential geometry. Let M be a locallyhomogeneous affine surface. The dimension of the space of germs of affine Killingvector fields (resp. affine gradient Ricci solitons) is constant on M . Let X i (resp. f i ) be affine Killing vector fields (resp. define affine gradient Ricci solitons) whichare defined on a connected open subset O of M . If there is a non-empty subset O ⊂ M with X = X (resp. f = f ) on O , then X = X (resp. f = f ) on O . Thus questions of passing from the local to the global for either affine Killingvector fields or affine gradient Ricci solitons involve the holonomy action of thefundamental group; there is no obstruction if M is assumed simply connected. Weshall not belabor the point and ignore the question of passing from local to globalhenceforth.1.6. Moduli spaces.
The moduli space Z A of isomorphism classes of germs ofType A structures is 2-dimensional [13]. The strata of Z A where Rank { ρ } = 1is handled by Theorem 3.8; it contains two components isomorphic to [0 , ∞ ) and OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 5 ( −∞ , Z A where ρ is non-degenerate of signature ( p, q ); these may be identified with closed simply connectedsubsets of R . Let Z B be the moduli space of Type- B structures. The strata of Z B where dim { K ( M ) } = 4 is handled by Theorem 3.8 since all these surfaces are also ofType A . We will also show in [4] that the strata of Z B where 2 ≤ dim { K ( M ) } ≤ Homogeneous affine surfaces
In this section, we use the structure groups described above acting on the families F A and F B to perform certain normalizations. Recall that a k -tensor T is said tobe symmetric if T ( v , . . . , v k ) = T ( v σ (1) , . . . , v σ ( k ) ) for every permutation σ , andthat T is said to be recurrent if ∇ T = ω ⊗ T for some 1-form ω .2.1. Rank 2 symmetric Ricci tensor.
We will show presently that ρ = ρ s if M is Type A . However, ρ need not be symmetric if M is Type B . Lemma 2.1. (1)
Let
M ∈ F A satisfy Rank { ρ } = 2 . Then ρ determines a flat pseudo-Riemannian metric on M . (2) Let
M ∈ F B satisfy Rank { ρ s } = 2 . (a) ρ s defines a pseudo-Riemannian metric of constant Gauss curvature κ . (b) κ = 0 if and only if ρ = 0 . (c) If ∇ is projectively flat, then the metric defined by ρ s has κ = 0 .Proof. If M is Type A , then ρ is symmetric. If Rank { ρ s } = 2, then ρ s defines apseudo-Riemannian metric. If M ∈ F A , then ρ s is invariant under the translationgroup ( a, b ) : ( x , x ) → ( x + a, x + b ). This group acts transitively on R andhence the components of ρ are constant. This implies ρ is flat. If M ∈ F B , then ρ s is invariant under the ax + b group ( a, b ) : ( x , x ) → ( ax , ax + b ). Thisnon-Abelian 2-dimensional Lie group acts transitively on R + × R and hence ρ s hasconstant Gauss curvature κ . This proves Assertion 1 and Assertion 2a. The proofof the remaining assertions follows as in [5]. (cid:3) Type A homogeneous affine surfaces. We omit the proof of the followingresult as it is a direct computation (see also [5]):
Lemma 2.2.
Let
M ∈ F A . Then (1) The Ricci tensor of M is symmetric ( ρ = ρ ) and one has: ρ = (Γ − Γ )Γ + Γ (Γ − Γ ) , ρ = Γ Γ − Γ Γ , ρ = − (Γ ) + Γ Γ + (Γ − Γ )Γ . (2) ∇ ρ is symmetric ( ρ = ρ = ρ , ρ = ρ = ρ ) and one has: ρ = 2 {− (Γ ) Γ + Γ (Γ (Γ − Γ ) + (Γ ) )+Γ (Γ Γ − Γ Γ ) } , ρ = 2 (cid:0) Γ (cid:0) (Γ ) − Γ Γ + Γ Γ (cid:1) − Γ Γ Γ (cid:1) , ρ = 2 (cid:0) Γ ( − Γ Γ − Γ Γ + Γ Γ ) + Γ Γ Γ (cid:1) , ρ = 2 { Γ (Γ (Γ − Γ ) + Γ Γ ) + (Γ ) Γ , − Γ (Γ Γ + (Γ ) ) } . If M ∈ F A , then we can always make a linear change of coordinates to replace M by an isomorphic surface where ρ = ρ dx ⊗ dx , i.e. ρ = ρ = ρ = 0.The following is a useful technical result: M. BROZOS-V´AZQUEZ E. GARC´IA-R´IO, AND P. GILKEY
Lemma 2.3.
Let
M ∈ F A . The following conditions are equivalent: (1) ρ ( M ) = ρ dx ⊗ dx . (2) Γ = 0 and Γ = 0 . (3) ρ = { Γ (Γ − Γ ) + Γ Γ } dx ⊗ dx .Proof. We assume Assertion 1 holds and apply Lemma 2.2.(1) Suppose first Γ is non-zero. By rescaling, we may suppose Γ = 1. Toensure ρ = 0, we set Γ = Γ Γ and obtain ρ = Γ ρ . Since ρ = 0, Γ = 0 and hence Γ = 0 as well.(2) Suppose next that Γ = 0. Setting ρ = 0 yields Γ Γ = 0. Since ρ = Γ (Γ − Γ ), Γ = 0. Thus Γ = 0. We now computethat ρ = Γ (Γ − Γ ) and ρ = Γ (Γ − Γ ). Consequently,Γ = 0.Thus in either eventuality we obtain Γ = 0 and Γ = 0 so Assertion 1 impliesAssertion 2. The proof that Assertion 2 implies Assertion 3 is a direct computation.The proof that Assertion 3 implies Assertion 1 is immediate. (cid:3) Definition 2.4.
Let
M ∈ F A . Choose X ∈ T P M so ρ ( X, X ) = 0 and set α X ( M ) := ∇ ρ ( X, X ; X ) · ρ ( X, X ) − and ǫ X ( M ) := Sign { ρ ( X, X ) } = ± . Lemma 2.5.
Let
M ∈ F A . (1) There exists a -form ω so ∇ k ρ = ( k + 1)! ω k ⊗ ρ for any k . (2) ρ is recurrent. (3) Ker { ρ } is a parallel distribution. (4) α X ( M ) and ǫ X ( M ) are independent of the choice of X and determineinvariants we will denote by α ( M ) and ǫ ( M ) .Proof. Choose coordinates on R so that ρ = ρ dx ⊗ dx . Assertion 1 then followsfrom Lemma 2.2 and Lemma 2.3, and Assertion 2 then follows from Assertion 1. Wehave ker( ρ ) = Span { ∂ } . Lemma 2.3 then shows Span { ∂ } is a parallel distributionas desired. Use Assertion 1 to express ρ = c ω ⊗ ω and ∇ ρ = c ω ⊗ ω ⊗ ω .One verifies α X ( M ) = ( ω ( X ) c ) ( ω ( X ) c ) − and ǫ X ( M ) = Sign { ω ( X ) c } areindependent of X . (cid:3) Remark 2.6.
Clearly α ( M ) = 0 if and only if M is symmetric. Furthermore, if α ( M ) = 0, then ǫ ( M ) = Sign( α ( M )) so ǫ is determined by α except in the sym-metric setting. We will show subsequently in Theorem 3.8 that α and ǫ determinethe local isomorphism class of a Type A surface with Rank { ρ } = 1.2.3. Type B homogeneous affine surfaces. We begin by extending Lemma 2.2to this setting. We omit the proof of the following result as it is a direct computa-tion (see [5]).
Lemma 2.7.
Let
M ∈ F B so Γ = ( x ) − C . (1) ρ = ( x ) − { C ( C − C + 1) + C ( C − C ) } . (2) ρ = ( x ) − {− C C + C C + C } . (3) ρ = ( x ) − {− C C + C C − C } . (4) ρ = ( x ) − { C C − ( C ) + C C − C C − C } . We use the coordinate transformation ( x , x ) → ( x , εx + x ) to partiallynormalize the Christoffel symbols. The following result will be used in the proof ofLemma 3.15 subsequently. Lemma 2.8.
Let
M ∈ F B . (1) If C = 0 , then by replacing x by x − εx , we may assume that C = 0 . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 7 (2) If C = 0 , C = 0 , C = 0 , and C − C = 0 , then by replacing x by x − εx , we may assume that C = 0 without changing the otherChristoffel symbols.Proof. Let ( u , u ) := ( x , εx + x ). We then have: du = dx , du = εdx + dx , ∂ u = ∂ x − ε∂ x , ∂ u = ∂ x , ∇ ∂ u ∂ u = ∇ ∂ x − ε∂ x ∂ x = ( x Γ − x Γ ε ) ∂ x + ⋆∂ x = ( x Γ − x Γ ε ) ∂ u + ⋆∂ u , u C = x C − ε · x C . We prove Assertion 1 by taking ε = x C ( x C ) − .Assume C = 0, C = 0, C = 0, and C − C = 0. We compute ∇ ∂ u ∂ u = x Γ ∂ x + x Γ ∂ − ǫ · x Γ ∂ x − ǫ · x Γ ∂ x + ǫ · x Γ ∂ x + ǫ · x Γ ∂ x = x Γ ∂ x + x Γ ∂ x − ǫ · x Γ ∂ x = x Γ ( ∂ x − ǫ∂ x ) + ( x Γ + ǫ { x Γ − · x Γ } ) ∂ x , ∇ ∂ u ∂ u = x Γ ∂ x + x Γ ∂ x − ǫ · x Γ ∂ x − ǫ · x Γ ∂ x = x Γ ∂ x , ∇ ∂ u ∂ u = x Γ ∂ x + x Γ ∂ x = 0, u Γ = x Γ , u Γ = x Γ + ǫ ( x Γ − · x Γ ), u Γ = 0, u Γ = x Γ , u Γ = 0 , u Γ = 0.We set ǫ = − ( x Γ − x Γ ) − · x Γ to establish Assertion 2. (cid:3) Remark 2.9.
We apply Lemma 2.8 to simplify the expressions of the Ricci tensor:(1) If C = 0 we may assume that C = 0 and express ρ = ( x ) − { C ( C − C + 1) + C C } , ρ = ( x ) − {− C C + C } , ρ = ( x ) − {− C C } , ρ = ( x ) − { ( C − C − C } .(2) If C = 0, C = 0, C = 0, and C − C = 0, we may assumethat C = 0 and express: ρ = ( x ) − { C ( C − C + 1) } , ρ = 0 , ρ = 0 , ρ = 0.The Ricci tensor of a Type B surface is not symmetric in general. Indeed, itis symmetric if and only if Γ = − Γ . Consequently, this family of surfaces isnot projectively flat in general, in contrast to Type A surfaces. We decompose theRicci tensor into its symmetric and its alternating parts in the form ρ = ρ s + ρ a . IfRank { ρ s } = 0, then ρ = ρ a (i.e., ρ ij = − ρ ji for all 1 ≤ i, j ≤ B surfacesand it is appropriate to introduce the necessary notation then. We now examinethe case that ρ is symmetric, ∇ ρ is symmetric, and Rank { ρ s } = 1. Lemma 2.10.
Let
M ∈ F B . The following conditions are equivalent: (1) We have that C = 0 , C = 0 , and C = 0 . (2) We have that (a) ρ = ( x ) − (1 + C − C ) C dx ⊗ dx . (b) ∇ ρ = ( x ) − ( − C )(1 + C − C ) C ) dx ⊗ dx ⊗ dx . (c) α ( M ) := ρ /ρ = 4(1 + C ) / { (1 + C − C ) C } . (3) ρ is symmetric, recurrent, and of rank and ∇ ρ is symmetric. M. BROZOS-V´AZQUEZ E. GARC´IA-R´IO, AND P. GILKEY
Proof.
A direct computation shows that Assertion 1 implies Assertion 2. It isimmediate that Assertion 2 implies Assertion 3. Assume Assertion 3 holds so ρ issymmetric. This implies C + C = 0. Since ρ is symmetric and has rank 1, wemay express ρ = ( x ) − ε ( a dx + a dx ) ⊗ ( a dx + a dx )where ε = ±
1. There are two possibilities.
Case 1: a = 0. By making the linear change of coordinates ˜ x = a x + a x , weobtain a new Type B surface with ρ = ( x ) − εdx ⊗ dx . Since ∇ ρ is recurrent, wehave ∇ ρ = ( x ) − ω ⊗ dx ⊗ dx . Since ∇ ρ is symmetric, we have ω = c ⊗ dx forsome constant c . Thus the only non-zero component of ∇ ρ is ρ . We compute ∇ ∂ ρ = − ( x ) − ε { C ) dx ⊗ dx + C dx ⊗ dx + C dx ⊗ dx } , ∇ ∂ ρ = − ( x ) − ε { C dx ⊗ dx + C dx ⊗ dx + C dx ⊗ dx } .This implies C = − C = 0 which is not possible. Case 2: a = 0 . We have ρ = ( x ) − ̺dx ⊗ dx for ̺ = 0. Then ∇ ∂ ρ = ⋆dx ⊗ dx − ̺ ( x ) − C ( dx ⊗ dx + dx ⊗ dx ) } , ∇ ∂ ρ = − ̺ ( x ) − { C dx ⊗ dx + C ( dx ⊗ dx + dx ⊗ dx ) } . Since ρ is recurrent, C = 0 and C = 0. As ρ is symmetric, C = − C = 0.Thus we obtain the relations of Assertion 1. (cid:3) If ρ = ρ dx ⊗ dx , then ker( ρ ) = Span { ∂ } . If, moreover, C = 0, C = 0,and C = 0, then ker( ρ ) is a parallel distribution which is totally geodesic.3. Affine Killing vector fields If X is a smooth vector field on M , let Φ Xt be the local flow defined by X . Werefer to Kobayashi-Nomizu [10, Chapter VI] for the proof of the following result. Lemma 3.1.
Let M = ( M, ∇ ) be an affine surface. (1) The following 3 conditions are equivalent and if any is satisfied, X is saidto be an affine Killing vector field : (a) (Φ Xt ) ∗ ◦ ∇ = ∇ ◦ (Φ Xt ) ∗ on the appropriate domain. (b) The Lie derivative L X ( ∇ ) of ∇ vanishes. (c) [ X, ∇ Y Z ] − ∇ Y [ X, Z ] − ∇ [ X,Y ] Z = 0 for all Y, Z ∈ C ∞ ( T M ) . (2) Let K ( M ) be the set of affine Killing vector fields. The Lie bracket gives K ( M ) the structure of a real Lie algebra. Furthermore, if X ∈ K ( M ) , if X ( P ) = 0 , and if ∇ X ( P ) = 0 , then X ≡ . (3) If M is an affine surface, then dim { K ( M ) } ≤ ; equality holds if and onlyif M is flat. The relations of Assertion 1c will be called
Killing equations . The followingresult characterizes Types A and B homogeneous affine surfaces by means of theLie Algebra structure of their affine Killing vector fields. It is a restatement ofArias-Marco and Kowalski [2, Lemma 1 and Lemma 2]. Lemma 3.2.
Let M = ( M, ∇ ) be a homogeneous affine surface. (1) M is of Type A if and only if there exists an Abelian sub-algebra g of K ( M ) of rank 2, i.e. there exist X, Y ∈ K ( M ) which are linearly independent atsome point P of M so that [ X, Y ] = 0 . (2) M is of Type B if and only if there exists a non-Abelian sub-algebra g of K ( M ) of rank 2, i.e there exist X, Y ∈ K ( M ) which are linearly independentat some point P of M so that [ X, Y ] = Y . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 9 If P is a point of a locally homogeneous surface M , let K P ( M ) be the Liealgebra of germs of affine Killing vector fields at P . If M is both Type A andType B , then there is a 2-dimensional Abelian Lie sub-algebra of K P ( M ) andthere is also a 2-dimensional non-Abelian Lie sub-algebra of K P ( M ). Consequentlydim { K P ( M ) } > Affine Killing vector fields on Type A surfaces. We begin by establishingsome technical results. Let ℜ ( · ) and ℑ ( · ) denote the real and imaginary parts of acomplex valued function. Let K A := Span { ∂ , ∂ } . If M ∈ F A , then Remark 1.3 shows K A ⊂ K ( M ). The adjoint action of K A makes K ( M ) into a K A module. This module action will play an important role in theproof of the following result. Lemma 3.3.
Let
M ∈ F A . Suppose that dim { K ( M ) } > . There exists a linearchange of coordinates so that M has the following properties: (1) There exists X ∈ K ( M ) so that one of the following possibilities holds: (a) X = ℜ{ e a x + a x } ∂ for = ( a , a ) ∈ C . (b) X = ( a x + a x ) ∂ for = ( a , a ) ∈ R . (2) ρ = ρ dx ⊗ dx . (3) If X ∈ K ( M ) , then X = ζ ( x , x ) ∂ + c ∂ for c ∈ R .Proof. We proceed seriatim.
Step 1: the proof of Assertion 1.
We complexify to set L := K ( M ) ⊗ R C and L A := K A ⊗ R C . Since dim { K ( M ) } >
2, we may choose X ∈ L − L . By Lemma 3.1, dim { L } ≤ ∂ i X = [ ∂ i , X ] ∈ L for i = 1 ,
2, there must be minimal non-trivial dependencerelations: ∂ r X + c r − ∂ r − X + · · · + c X = 0 with r > ,∂ s X + ˜ c s − ∂ s − X + · · · + ˜ c X = 0 with s > c i and ˜ c i . We factor the associated characteristicpolynomials to express these dependence relation in the form: u Y t =1 ( ∂ − λ t ) µ t X = 0 for λ t ∈ C distinct and µ t ≥ , (3.a) w Y v =1 ( ∂ − η v ) ν v X = 0 for η v ∈ C distinct and ν v ≥ . (3.b) Case 1.
Suppose that some λ t is non-zero (if all λ t are zero and some η v is non-zerothe analysis is analogous). By reordering the roots, we may assume λ = 0. Sincewe have chosen a minimal dependence relation, we have0 = Y := ( ∂ − λ ) µ − . . . ( ∂ − λ u ) µ u X ∈ L . By replacing X by Y , we may assume the dependence relation of Equation (3.a) is( ∂ − λ ) X = 0. This implies X = e λ x ( ξ ( x ) ∂ + ξ ( x ) ∂ ) ∈ L where λ = 0 . A similar argument shows we may assume that Equation (3.b) takes the form( ∂ − η ) X = 0 for some η (possibly 0). We then conclude X = e λ x + η x X ∈ L for 0 = X ∈ L A and 0 = λ . If ℜ ( X ) and ℑ ( X ) are linearly dependent over R , then we can multiply X byan appropriate non-zero complex number to assume 0 = X ∈ K A is real. Thisimplies ℜ ( X ) = ℜ{ e λ x + η x } X has the form given in Assertion 1a. We thereforesuppose that ℜ ( X ) and ℑ ( X ) are linearly independent. We can make a linearchange of coordinates to assume X = ( ∂ − √− ∂ ) = ∂ z where z = x + √− x .Thus Z = e φ ∂ z ∈ L for some suitably chosen non-trivial linear function φ . Case 1a.
Suppose φ is purely imaginary. This implies φ = √− a x + a x ) for0 = ( a , a ) ∈ R . We can rotate R and then rescale to suppose that φ = √− x .We then have X = { cos( x ) + √− x ) } ( ∂ − √− ∂ ) / ℜ{ X } = (cos( x ) ∂ + sin( x ) ∂ ) ∈ K ( M ) . We have Killing equations:(1 − ) cos( x ) + Γ sin( x ) = 0 , (Γ − ) cos( x ) + (1 + 2Γ ) sin( x ) = 0 , Γ cos( x ) = 0 , (Γ − Γ ) cos( x ) + Γ sin( x ) = 0 . We solve these relations to seeΓ = 0 , Γ = − , Γ = , Γ = 0 , Γ = 0 , Γ = . The Ricci tensor of this structure is zero; this is false as M is assumed non-flat. Case 1b.
Suppose φ is holomorphic. We can then rotate and rescale to ensurethat φ ( z ) = z so ℜ ( X ) = ℜ{ e x + √− x ( ∂ − √− ∂ ) / } = e x (cos( x ) ∂ + sin( x ) ∂ ) / . We have Killing equations:(1 + Γ ) cos( x ) + (Γ + 2Γ ) sin( x ) = 0 , Γ cos( x ) + (1 − Γ + 2Γ ) sin( x ) = 0 , Γ cos( x ) − (1 + 2Γ + Γ ) sin( x ) = 0 . We solve these equations to seeΓ = − , Γ = 0 , Γ = 0 , Γ = − , Γ = 1 , Γ = 0 . The Ricci tensor of this structure is zero; this is false as M is assumed non-flat. Case 1c.
Assume that φ is not purely imaginary and that φ is not holomorphic.Since ¯ X ∈ L ,[ X, ¯ X ] = e φ + ¯ φ { ∂ z ¯ φ · ∂ ¯ z − ∂ ¯ z φ · ∂ z } = 2 √− e φ + ¯ φ ℑ{ ∂ z ¯ φ · ∂ ¯ z } ∈ L . Since φ is not purely imaginary, the exponent φ + ¯ φ = a x + a x is non-trivialand real. Since φ is not holomorphic, 0 = ξ := ℑ{ ∂ z ¯ φ · ∂ ¯ z } ∈ K A . We can changecoordinates to assume ξ = ∂ . We then have −√− Z, ¯ Z ] = e ˜ a x +˜ a x ∂ satisfiesthe hypotheses of Assertion 1a. This completes the analysis of Case 1. Case 2.
Neither dependence relation involves a complex root of the characteristicpolynomials, i.e. we have ∂ r X = ∂ s X = 0. Since X / ∈ K A , ( r, s ) = (1 , r > X by ∂ r − X to ensure r ≤
2. We then argue similarly to choose X so s ≤ X = X i =0 1 X j =0 ( x ) i ( x ) j X ij for X ij ∈ K A . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 11 If X is non-zero, we can apply ∂ to reduce the order and after subtracting theconstant term obtain an element with the form given in Assertion 1b. Otherwise,we may simply subtract X to see that there exists X ∈ K ( M ) so that X = a ji x i ∂ j ∈ K ( M ) for ( a ji ) = 0 . If Rank { ( a ji ) } = 1, then we can change coordinates to assume X has the formgiven in Assertion 1b. We therefore assume Rank { ( a ji ) } = 2 and argue, at length,for a contradiction. Only the Jordan normal form of the coefficient matrix ( a ji ) isrelevant since we are working modulo linear changes of coordinates. Furthermore,we can always rescale X as needed. Case 2a. A is diagonalizable. We may suppose X = x ∂ + ax ∂ for a = 0. Weobtain the equations:Γ = 0 , ( a − = 0 , a Γ = 0 , Γ = 0 , ( − a )Γ = 0 , a Γ = 0 . Thus the only possibly non-zero Christoffel symbols are Γ and Γ . Since itis not possible that ( a −
2) = 0 and ( − a ) = 0 simultaneously, we also haveΓ Γ = 0. This implies ρ = 0 so this case is ruled out. Case 2b. A has two equal non-zero eigenvalues and non trivial Jordan normalform. We may suppose that X = ( x + x ) ∂ + x ∂ and obtain Killing equations:Γ − Γ = 0 , Γ = 0 , Γ + Γ − Γ = 0 , Γ + Γ = 0 , + Γ − Γ = 0 , + Γ = 0 . We solve these equations to see Γ = 0 and hence ρ = 0 so this case is ruled out. Case 2c.
The matrix A has two complex eigenvalues with non-zero imaginary part.We may assume X = ( ax + x ) ∂ + ( − x + ax ) ∂ is an affine Killing vector fieldfor some a ∈ R . We use the Killing equations to eliminate variables recursively.We set Γ = 2 s and Γ = 2 t . At each stage we simplify the resulting Killingequations based on the previous computations:(1) The Killing equation 4 s + 2 at + Γ = 0 yields Γ = − s − at .(2) The Killing equation 2 as + t + a t − Γ = 0 yields Γ = 2 as + t + a t .(3) The Killing equation 4 as − t + a t + Γ = 0 yields Γ = − as + t − a t .(4) The Killing equation 2(1 + a ) s + 3 at + a t + Γ = 0 yieldsΓ = − a ) s − at − a t .We now obtain Killing equations in the parameters ( s, t ) which imply 3 s + at = 0and 2 as + t (3 + a ) = 0. We set s = − at/ t + a t/ t = 0 so s = 0 and Γ = 0. Thus this case is ruled out. This completesthe proof of Assertion 1. Step 2: the proof of Assertion 2.
By Assertion 1, X = f ( x , x ) ∂ ∈ K ( M ) forsome non-constant function f . Choose P ∈ R so df ( P ) = 0. Let Y = c ∂ + c ∂ for ( c , c ) = (0 , {L X ( ρ ) } ( Y, Y ) = X ( ρ ( Y, Y )) − ρ ([ X, Y ] , Y ) . Because ρ ( Y, Y ) is constant, X { ρ ( Y, Y ) } = 0. For generic ( c , c ),[ X, Y ]( P ) = { Y ( f )( P ) } ∂ = 0 so ρ ( ∂ , Y ) = 0 . This implies ρ = ρ = 0 so ρ = ρ dx ⊗ dx . Step 3: the proof of Assertion 3.
Let X = ξ ( x , x ) ∂ + ξ ( x , x ) ∂ ∈ K ( M ).Let Y = c ∂ + c ∂ . We argue as above to see that ρ ([ X, Y ] , Y ) = 0. Since ρ = ρ dx ⊗ dx , this implies c ρ ( c ∂ ξ + c ∂ ξ ) = 0 for all ( c , c ) ∈ R . This implies ξ is constant which establishes Assertion 3 and completes the proofof the Lemma. (cid:3) Lemma 3.3 focuses attention on the case that Rank { ρ } = 1. The following resultrelates the rank of the Ricci tensor with the dimension of the space of affine Killingvector fields. Theorem 3.4.
Let
M ∈ F A . (1) Suppose ρ = ρ dx ⊗ dx . (a) If Γ = 0 , then X ∈ K ( M ) if and only if X = e − Γ x ξ ( x ) ∂ + X for X ∈ K A where ξ satisfies ξ ′′ + (2Γ − Γ ) ξ ′ + Γ Γ ξ = 0 . (b) If Γ = 0 , then X ∈ K ( M ) if and only if X = ( ξ ( x ) + c x ) ∂ + X for X ∈ K A where ξ satisfies ξ ′′ + (2Γ − Γ ) ξ ′ − c Γ = 0 . (2) The following assertions are equivalent. (a) dim { K ( M ) } = 4 . (b) dim { K ( M ) } > . (c) Rank { ρ } = 1 . (3) The following assertions are equivalent: (a) dim { K ( M ) } = 2 . (b) Rank { ρ } = 2 .Proof. We use Lemma 2.3 to impose the conditions Γ = 0 and Γ = 0 andLemma 3.3 to write X = ζ ( x , x ) ∂ + c ∂ . The Killing equations now become ζ (2 , + Γ ζ (1 , = 0 ,ζ (1 , + Γ ζ (0 , = 0 ,ζ (0 , − Γ ζ (1 , + (2Γ − Γ ) ζ (0 , = 0 . We establish Assertion 1 by examiningg cases.(1) Suppose that Γ = 0. We have ζ ( x , x ) = u ( x ) + u ( x ) e − Γ x . AKilling equation is Γ u ′ = 0. Thus we may take u constant and delete itfrom further consideration. The remaining Killing equation is the conditionof Assertion 1a.(2) Suppose Γ = 0. We have ζ ( x , x ) = u ( x ) + u ( x ) x . A Killingequation is u ′ = 0 and hence u ( x ) = c is constant. The remainingKilling equation is the condition of Assertion 1b.Clearly Assertion 2a implies Assertion 2b. We use Lemma 3.3 to see that Asser-tion 2b implies Assertion 2c. We will apply Assertion 1 to see Assertion 2c impliesAssertion 2a. We argue as follows. Suppose first Γ = 0. Let { ξ , ξ } be a basisfor the space of solutions to the Equation of Assertion 1a. Then K ( M ) = Span R { ξ ( x ) e − Γ x ∂ , ξ ( x ) e − Γ x ∂ , ∂ , ∂ } and hence dim { K ( M ) } = 4. Suppose on the other hand that Γ = 0. Choose asolution ξ ( x ) to the Equation of Assertion 1b with c = 1, i.e. we have ζ ( x , x ) = x + ξ ( x ) where ξ ′′ + (2Γ − Γ ) ξ ′ − Γ = 0 . Let { ξ , ξ } be a basis for the space of solutions to the homogeneous equation ξ ′′ + (2Γ − Γ ) ξ ′ = 0. Then K ( M ) = Span R { ξ ∂ , ξ ∂ , ξ ∂ , ∂ , ∂ } . Since we may take ξ = 1, ξ ∂ = ∂ and we see that dim { K ( M ) } = 4. Thiscompletes the proof of Assertion 2; the final Assertion is now immediate. (cid:3) There are several Lie algebras which will play an important role in our analysis.Let A := Span R { e , e } with Lie bracket [ e , e ] = e ; up to isomorphism, A isthe only non-trivial real Lie algebra of dimension two; it is the Lie algebra of the OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 13 “ ax + b ” group. We adopt the notation of Patera et. al [16] to define several otherLie algebras. Let { e , e , e , e } be a basis of R . We define the following solvableLie algebras by specifying their bracket relations. • A ⊕ A : the relations of the bracket are given by[ e , e ] = e , [ e , e ] = e . • A b , : the relations of the bracket for − ≤ b ≤ e , e ] = e , [ e , e ] = (1 + b ) e , [ e , e ] = e , [ e , e ] = be . • A , : the relations of the bracket are given by[ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e . Definition 3.5.
Let M ⋆⋆ be the affine surface defined by the structures: M : Γ = −
1, Γ = 0, Γ = 1, Γ = 0, Γ = 0, Γ = 2. M c : Γ = −
1, Γ = 0, Γ = c , Γ = 0, Γ = 0, Γ = 1 + 2 c ,where c + c = 0. M c : Γ = 0, Γ = 0, Γ = c , Γ = 0, Γ = 0, Γ = 1 + 2 c ,where c + c = 0. M c : Γ = 0, Γ = 0, Γ = 1, Γ = 0, Γ = c , Γ = 2. M c : Γ = −
1, Γ = 0, Γ = c , Γ = 0, Γ = −
1, Γ = 2 c .We use Lemma 2.3 to see ρ ( M ⋆⋆ ) = ρ dx ⊗ dx = 0 for ρ = 0 so none of theseexamples is flat. We compute ρ and α : ρ ( M ) = 1 , α ( M ) = 16 ,ρ ( M c ) = c + c, α ( M c ) = c ) c + c ∈ ( −∞ , ∪ (16 , ∞ ) ,ρ ( M c ) = c + c, α ( M c ) = c ) c + c ∈ ( −∞ , ∪ (16 , ∞ ) ,ρ ( M c ) = 1 , α ( M c ) = 16 ,ρ ( M c ) = 1 + c , α ( M c ) = c c ∈ [0 , . The general linear group GL(2 , R ) acts on the space of Christoffel symbols by pull-back; we say that Γ and Γ are linearly equivalent if there exists T ∈ GL(2 , R ) sothat T ∗ ( ∇ Γ ) = ∇ Γ . We have the following classification result. Lemma 3.6. (1) If M ∈ F A and Rank( ρ ) = 1 , then M is linearly equivalent to M , M c , M c , M c , or M c . (2) K ( M ) = Span R { e x ∂ , x e x ∂ } ⊕ K A ≈ A , . (3) K ( M c ) = Span R { e x ∂ , e x + x ∂ } ⊕ K A ≈ A ⊕ A . (4) K ( M c ) = Span { e x ∂ , x ∂ } ⊕ K A ≈ A ⊕ A . (5) K ( M c ) = Span { x ∂ , ( c · ( x ) + 2 x ) ∂ } ⊕ K A ≈ A , . (6) K ( M c ) = Span R { e x cos( x ) ∂ , e x sin( x ) ∂ } ⊕ K A ≈ A , . (7) M , M c , M c , and M c are also Type B ; M c is not Type B .Proof. Assume ρ has rank 1 and make a linear change of coordinates to assume ρ = ρ dx ⊗ dx . By Theorem 3.4, there exists X ∈ K ( M ) − K A of the form ζ ( x , x ) ∂ where ζ is non-constant. Lemma 3.3 then shows either ζ = e a x + a x for (0 , = ( a , a ) ∈ R (Case 1 and Case 2 below), or ζ = a x + a x for(0 , = ( a , a ) ∈ R (Case 3 below), or ζ = ℜ{ e a x + a x } for ( a , a ) ∈ C − R (Case 4 below). We examine these possibilities seriatim. Case 1.
Assume e a x + a x ∂ ∈ K ( M ) for a = 0. Let( u , u ) := ( a x + a x , x ) so e u ∂ u = ( a ) − e a x + a x ∂ ∈ K ( M ) . Thus we may assume that e x ∂ ∈ K ( M ). Let X := ∂ + ∂ and X := e x ∂ , then { X ( P ) , X ( P ) } are linearly independent for any point P ∈ R . By Lemma 3.2, M is also Type B since [ X , X ] = X . A direct computation shows X = e x ∂ isan affine Killing vector field if and only ifΓ = − , Γ = 0 , Γ = 0 , Γ = 0 . We impose these relations and obtain ρ = Γ (Γ − Γ ) = 0. Two sub-casespresent themselves when we search for another affine Killing vector field: Case 1a.
Assume Γ = 2Γ . We set Y = x e x ∂ and verify Y is an affineKilling vector field. By Theorem 3.4, dim { K ( M ) } = 4. Thus K ( M ) = Span R { e x ∂ , x e x ∂ } ⊕ K A . Since ρ = 0, Γ = 0. By rescaling x , we may assume that Γ = 1; this yieldsthe surface M . Set e := e x ∂ , e := x e x ∂ , e := − ∂ , e := − ∂ . We thenhave the bracket relations of the Lie algebra A , :[ e , e ] = e , [ e , e ] = e , [ e , e ] = e . Case 1b.
Assume Γ − = 0. Then Y = e x +(Γ − ) x ∂ is an affineKilling vector field distinct from e x ∂ . By replacing x by (Γ − ) − x , wemay assume Y = e x + x ∂ ; the Killing equations then yield Γ = 2Γ + 1 andthus K ( M ) = Span R { e x ∂ , e x + x ∂ } ⊕ K A . We set e := ∂ , e := e x + x ∂ , e := ∂ − ∂ , e := e x ∂ . This yields the surface M c . We then have the bracket relations of the Lie algebra A ⊕ A :[ e , e ] = e , [ e , e ] = e . Case 2.
Assume e a x + a x ∂ ∈ K ( M ) for a = 0. Hence e a x ∂ ∈ K ( M ) for a = 0. We may rescale x to assume a = 1. A direct computation shows e x ∂ isan affine Killing vector field if and only ifΓ = 0 , Γ = 0 , Γ = 0 , Γ = 1 + 2Γ . If we set Y = ( x − Γ x ) ∂ , then this is an affine Killing vector field. We maymake a linear change of variables to replace x by x − Γ x to obtain x ∂ is anaffine Killing vector field; this implies Γ = 0. This yields the surface M c . Wethen have K ( M ) = Span { e x ∂ , x ∂ } ⊕ K A . We set X = ∂ and X = e x ∂ . Since [ X , X ] = X and { X ( P ) , X ( P ) } arelinearly independent for any P ∈ R , Lemma 3.2 implies M is Type B as well. Weset e := − x ∂ − ∂ , e := − ∂ , e := ∂ , and e := e x ∂ . We then have thebracket relations of the Lie algebra A ⊕ A :[ e , e ] = e , [ e , e ] = e . Case 3.
Assume ( a x + a x ) ∂ ∈ K ( M ) for ( a , a ) = (0 , x and x to replace a i by λ i a i . Thus we need onlyconsider ( a , a ) ∈ { (1 , , (1 , , (0 , } . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 15
Case 3a.
Suppose ( a , a ) = (0 ,
1) so x ∂ ∈ K ( M ). A direct computation shows x ∂ is an affine Killing vector field if and only ifΓ = 0 , Γ = 0 , Γ = 0 , Γ = 2Γ . We then have ρ = (Γ ) . By rescaling x , we may assume Γ = 1 and henceΓ = 2. We obtain the surface M c . We set Y = (2 x + c · ( x ) ) ∂ and verifythat Y is an affine Killing vector field. Thus K ( M ) = Span R { x ∂ , (2 x + c · ( x ) ) ∂ } ⊕ K A . We set X = ∂ and X = ∂ +( x + c · ( x ) / ∂ . Then { X ( P ) , X ( P ) } are linearlyindependent for any point P ∈ R . Set e := ∂ , e := x ∂ , e := − ∂ + cx ∂ , e := ( c · ( x ) + x ) ∂ . We obtain the bracket relations of the Lie algebra A , :[ e , e ] = e , [ e , e ] = e , [ e , e ] = e . Case 3b.
Suppose ( a , a ) = (1 ,
0) so x ∂ ∈ K ( M ). The Killing equations yieldthe relations: Γ = 0 , Γ = 0 , Γ = 0 , Γ = 0 . Suppose first 2Γ = Γ . Set Y = e (Γ − ) x ∂ . We then verify that Y isan affine Killing vector field. Thus this is subsumed in Case 2. We may thereforesuppose 2Γ = Γ and we obtain that x ∂ also is an affine Killing vector field.This is subsumed in Case 3a. Case 3c.
Suppose ( a , a ) = (1 ,
1) so ( x + x ) ∂ ∈ K ( M ). By replacing x by x + x , we obtain x ∂ ∈ K ( M ). This is subsumed in Case 3b. Case 4.
Assume ( e α x + α x ) ∂ is a complex affine Killing vector field where wehave ( α , α ) ∈ C − R . Set α = a + √− a and α = b + √− b . Then thefollowing two vector fields are affine Killing vector fields: X := e a x + b x cos( a x + b x ) ∂ ,Y := e a x + b x sin( a x + b x ) ∂ . Consequently K ( M ) = Span R { X, Y } ⊕ K A . Case 4a.
Suppose a = 0. We can then make a linear change of coordinates toassume X = e a x + b x cos( x ) ∂ . The Killing equations yield:0 = ( − a + a Γ − b Γ ) cos( x ) − (2 a + Γ ) sin( x ) , ( a cos( x ) − sin( x )) . This implies: Γ = 0 , ( a ) + a Γ − , a + Γ = 0 . Thus Γ = − a . We show this case does not occur by deriving the contradiction:0 = ( a ) + a Γ − − − a . Case 4b.
Suppose a = 0 and normalize x so that b = 1 and X = e a x + b x cos( x ) ∂ . Suppose a = 0 so X = e b x cos( x ) ∂ . We obtain two relations: b + 2 b Γ − b Γ − − b − + Γ = 0.This implies b = (Γ − ) /
2. We derive a contradiction and show this casecan not occur by computing: b + 2 b Γ − b Γ − − (4 + 4(Γ ) − Γ + (Γ ) ) / − (4 + (2Γ − Γ ) ) / . Thus a = 0 so we can renormalize the coordinates to ensure X = e x cos( x ) ∂ .The bracket with ∂ then yields ˜ X = e x sin( x ) ∂ also is an affine Killing vectorfield. This generates the 4-dimensional Lie algebra K ( M ). Let e := e x cos( x ) ∂ , e := e x sin( x ) ∂ , e := − ∂ , e := − ∂ . We then have the bracket relations of A , :[ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e . This establishes Assertions 1-6; Assertion 7 follows from Lemma 3.2. (cid:3)
Remark 3.7.
No surface in one family of Definition 3.5 is linearly isomorphicto a surface in another family. We argue as follows to see this. The Lie algebra K ( M c ) is A , ; this is different from the Lie algebras of the other 4 families sothis family is distinct. Similarly, the Lie algebra of M or M c is A , while theLie algebra of M c or M c is A ⊕ A . So we must construct a linear invariantdistinguishing M from M c or distinguishing M c from M c . The Ricci tensor ofany surface in Definition 3.5 has rank 1 so ker( ρ ) is a 1-dimensional distribution;we have normalized the coordinate system so ker( ρ ) = ∂ · R . Let ρ := Γ ijj dx i .Since contraction of an upper against a lower index is invariant under the action ofGL(2 , R ), ρ and hence dim { ker( ρ ) ∩ ker( ρ ) } is a linear invariant. We compute ρ M ( ∂ ) = − , ρ M c ( ∂ ) = − , ρ M c ( ∂ ) = 0 , ρ M c ( ∂ ) = 0 . Thus ker( ρ ) ∩ ker( ρ ) = { } if M = M or M = M c while ker( ρ ) ∩ ker( ρ ) = { } if M = M c or M = M c . Thus in fact the 5 families of Definition 3.5 are distinctunder linear equivalence and Lemma 3.6 is minimal in this respect.Although the 5 basic families of Definition 3.5 are distinct under linear equiva-lence, there are non-linear changes of coordinates that can be used to relate mem-bers of different families. We use such changes to establish the following result thatshows that the invariants α and ǫ form a complete system of invariants for Type A surfaces where the Ricci tensor has rank 1. Theorem 3.8.
Let M and ˜ M be Type A affine surfaces with Rank { ρ } = 1 . As-sume that α ( M ) = α ( ˜ M ) = α and that ǫ ( M ) = ǫ ( ˜ M ) = ǫ . (1) If α = 16 , then M ≈ ˜ M , K ( M ) ≈ A , , and M is also of Type B . (2) If α ∈ (0 , , then M ≈ ˜ M , K ( M ) ≈ A , and M is not of Type B . (3) If α / ∈ [0 , , then M ≈ ˜ M , K ( M ) ≈ A ⊕ A , and M is also of Type B . (4) Assume α = 0 . (a) If ǫ < , then M ≈ ˜ M , K ( M ) ≈ A ⊕ A , and M is also of Type B . (b) If ǫ > , then M ≈ ˜ M , K ( M ) ≈ A , and M is not of Type B .Proof. We first deal with the surfaces M and M c .Assume x Γ = − x Γ = 0, x Γ = 0, and x Γ = 0. Set u = e − x and u = x . Then du = − e − x dx , du = dx ,∂ u = − e x ∂ , ∂ u = ∂ . We then have Span R { ∂ , e x ∂ , ∂ } = Span R {− u ∂ u , ∂ u , ∂ u } . We compute: ∇ ∂ u ∂ u = e x ∇ ∂ { e x ∂ } = e x { (1 + x Γ ) ∂ + x Γ ∂ } , ∇ ∂ u ∂ u = − e x ∇ ∂ ∂ = − e x { x Γ ∂ + x Γ ∂ } , ∇ ∂ u ∂ u = ∇ ∂ ∂ = x Γ ∂ + x Γ ∂ . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 17
This implies that: u Γ = − (1 + x Γ ) · e x = 0 , u Γ = x Γ ∈ R , u Γ = − x Γ · e − x = 0 , u Γ = x Γ · e x = 0 , u Γ = − x Γ · e x = 0 , u Γ = x Γ ∈ R . Thus α is unchanged and ˜ M := ( R + × R , u Γ) is isomorphic to M := ( R , x Γ).(This shows, incidentally, that ( R , x Γ) is incomplete in this instance).
Case 1. The surface M . We may identify M with M . We will discuss thesurfaces M c for more general c subsequently. Case 2. The surfaces M c . We may identify M c with M c . Let x = Γ . Wehave α = x ) x + x . This is symmetric about the line x = − . We note that α = 0precisely when Γ = − ; in this setting ρ < α = 0, then α takes values in ( −∞ , ∪ (16 , ∞ ). There are two possiblevalues of x (and two corresponding surfaces). We make a linear change of coordi-nates x → x − x and x → x to have K ( M ) = Span R { e x − x ∂ , e x + x ∂ }⊕ K A .We have e x ± x ∂ are affine Killing vector fields if and only if the following equationsare satisfied:0 = Γ − Γ + 1 , + Γ + 1 , , , − Γ + 1 , − Γ + 1 , + Γ , − Γ , − Γ − Γ + 1 , + Γ − Γ − , , , or equivalentlyΓ = − , Γ = 0 , Γ = 0 , Γ = 1 , Γ = 2Γ . We now have α = 16 x / ( x −
1) where x = Γ . The symmetry can now berealized by x → − x , i.e. Γ → − Γ . Thus α and ǫ completely detect thesurfaces M c . Case 3. The surface M c . These surfaces have been identified with the surfaces M c and dealt with in Case 2. Case 4. The surfaces M c . We have the relations x Γ = 0 , x Γ = 0 , x Γ = 1 , x Γ = 0 , x Γ = 2 . We have α = 16 and K ( M ) = Span R { x ∂ , ( x Γ ( x ) + 2 x ) ∂ } ⊕ K A . Theparameter c := x Γ is undetermined. Let u = x + x Γ ( x ) and u = x bea change of coordinates. We have du = dx + x Γ x dx , du = dx ,∂ u = ∂ , ∂ u = − x Γ x ∂ + ∂ . We compute: ∇ ∂ u ∂ u = ∇ ∂ ∂ = x Γ ∂ + x Γ ∂ = 0, ∇ ∂ u ∂ u = − x Γ x ( x Γ ∂ + x Γ ∂ ) + x Γ ∂ + x Γ ∂ = ∂ = ∂ u , ∇ ∂ u ∂ u = ( x Γ x ) ∇ ∂ ∂ − x Γ x ∇ ∂ ∂ − x Γ ∂ + ∇ ∂ ∂ = 0 − x Γ x ∂ − x Γ ∂ + x Γ ∂ + x Γ ∂ = 2 ∂ u . Consequently, u Γ = 0 , u Γ = 0 , u Γ = 1 , u Γ = 0 , u Γ = 0 , u Γ = 2 , so the surfaces M and M c are equivalent for any c . Case 5. The surfaces M c . We have the relationsΓ = − , Γ = 0 , Γ = 0 , Γ = − , Γ = 2Γ . We have α = 16 x / (1 + x ) takes values in [0 ,
16) where x = Γ . If α = 0,there is only one surface given by Γ = 0 and we have ρ = 1 + (Γ ) = 1corresponding to Theorem 3.8 (4b). If α ∈ (0 , ± Γ and the symmetry is realized by x → − x . We have M is not of Type B and K ( M ) ≈ A , , thus Assertion 2 follows.This shows that ( α, ǫ ) completely determines the isomorphism type of M andcompletes the proof of Theorem 3.8. (cid:3) We summarize our conclusions as follows:
Table 1.
Classification of homogeneous affine surfaces of Type A with Rank { ρ } = 1. Let κ ( M ) := dim { K ( M ) } . α ǫ M K ( M ) κ ( M ) Type A Type B α < − M c , M c , | c + | < A ⊕ A X X α = 0 − M c , M c , c = − A ⊕ A X X α = 0 +1 M A , X No0 < α <
16 +1 M c , c = 0 A , X No α = 16 +1 M , M c , c ∈ R A , X X < α +1 M c , M c , < | c + | A ⊕ A X X
Affine Killing vector fields on Type B homogeneous surfaces. Linearequivalence for Type A surfaces is the action of GL(2 , R ). Linear equivalence forType B surfaces is a bit more subtle in view of Remark 1.3. Lemma 3.9.
Let T b,c : ( x , x ) → ( x , bx + cx ) for c = 0 . Let C and ˜ C defineaffine manifolds M and ˜ M of Type B . Then M and ˜ M are linearly equivalent ifand only if there exists T b,c so T ∗ b,c C = ˜ C .Proof. Let G := { T : ( x , x ) → ( tx , ux + vx + w ) } for t > v = 0 be the4-dimensional subgroup of GL(2 , R ) which preserves R + × R . Then by definition, M is linearly equivalent to ˜ M if and only if there exists T ∈ G so that T ∗ C = ˜ C .There are two non-Abelian subgroups of G which play an important role. Set H := { S : ( x , x ) → ( ax , ax + b ) for a > } and I := { T b,c } . The subgroups H and I generate G as a Lie group. By Remark 1.3, H preservesType B structures. Thus only the action of I is relevant in studying linear equiva-lence for Type B structures and the Lemma follows. (cid:3) The Lie group I plays the crucial role in studying linear equivalence for Type B structures; the shear ( x , x ) → ( x , εx + x ) and the rescaling ( x , x ) → ( x , cx )for c = 0 generate I and will play a central role in what follows. The group H alsoplays an important role. Let K B be the Lie algebra of H . Then K B := Span { x ∂ + x ∂ , ∂ } ⊂ K ( M ) for any M ∈ F B . This non-Abelian Lie sub-algebra plays the same role in the analysis of Type B surfaces that K A played in the analysis of Type A surfaces. OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 19
Let su ( , ) be the Lie algebra of SU (1 ,
1) or, equivalently, of SL (2 , R ). It is theLie algebra on 3 generators (also denoted by A , in [16]) satisfying the relations:[ e , e ] = e , [ e , e ] = e , [ e , e ] = − e . (3.c) Definition 3.10.
For c ≥
0, set N ± := M ( C = − , C = 0, C = 0, C = − , C = ∓ , C = 0). N c := M ( C = − , C = 0, C = 1, C = − , C = c , C = 2). N := M ( C = − C = 0, C = 0, C = − C = − C = 0). N := M ( C = − C = 0, C = 0, C = − C = 1, C = 0).We show that these surfaces are not flat and thus N ⋆⋆ is Type B by computing: ρ ( N ± ) = ± ( x ) − dx ⊗ dx ,ρ ( N c ) = ( x ) − { ( dx ⊗ dx − dx ⊗ dx ) + (1 − c ) dx ⊗ dx } ,ρ ( N ) = ( x ) − ( − dx ⊗ dx + dx ⊗ dx ) ,ρ ( N ) = ( x ) − ( − dx ⊗ dx − dx ⊗ dx ) . (3.d)The main result of this section is the following: Theorem 3.11. If M ∈ F B , then ≤ dim { K ( M ) } ≤ . (1) If dim { K ( M ) } = 4 , then ρ = ( x ) − ˜ ρ dx ⊗ dx , C = C = C = 0 , M is also of Type A , and up to linear equivalence one of the following 3possibilities holds: (a) C − C = 0 , C = 1 , ˜ ρ = (1 + C ) C = 0 , and K ( M ) = Span { x ∂ − x log( x ) ∂ , x ∂ } ⊕ K B . (b) C = 0 , ˜ ρ = (1 + C − C ) C = 0 , and K ( M ) = Span { x ∂ , ( x ) a ∂ } ⊕ K B for some a = 0 . (c) C = 0 , ˜ ρ = ( C ) = 0 , and K ( M ) = Span { x ∂ , log( x ) ∂ } ⊕ K B . (2) If dim { K ( M ) } = 3 , then K ( M ) = Span { X ( σ ) } ⊕ K B ≈ su ( , ) where X ( σ ) := 2 x x ∂ + { ( x ) + σ · ( x ) } ∂ for σ ∈ {− , , } , M is not of Type A , and up to linear equivalence, one of the followingpossibilities holds: (a) σ = 0 , M = N ± , and M is not Type C . (b) σ = 0 , M = N c , and M is not Type C . (c) σ = 1 , M = N , and M is Type C . (d) σ = − , M = N , and M is Type C . (3) For each of the 3 structures listed in Assertion 1, dim { K ( M ) } = 4 . Foreach of the 4 structures listed in Assertion 2, dim { K ( M ) } = 3 . Remark 3.12. If M is Type C , then dim { K ( M ) } = 3 , ρ = ρ s , and Rank { ρ } = 2 .Thus if M ∈ F B then M is also of Type C if and only if either Assertion 2c orAssertion 2d of Theorem 3.11 holds. Similarly, M is both Type B and Type A ifand only if Assertion 1 of Theorem 3.11 holds. The proof of this result will occupy most of this section and will be a directconsequence of the following lemmas. It gives a complete description of thosehomogeneous affine surfaces of Type B with dim { K ( M ) } >
2. If X ∈ C ∞ ( T M ) isa smooth vector field on M , letΘ := ad( x ∂ + x ∂ ) i.e. Θ( X ) := [ x ∂ + x ∂ , X ] . (3.e) Lemma 3.13.
Let X ∈ C ∞ ( T M ) be polynomial in ( x , x ) . If X is homogeneousof degree ℓ , then Θ( X ) = ( ℓ − X . Proof.
Let X = X i + j = ℓ ( x ) i ( x ) j ( c i,j ∂ + c i,j ∂ ) for c νi,j ∈ R . Then:[ x ∂ , X ] = X i + j = ℓ ( x ) i ( x ) j { ( i − c i,j ∂ + ic i,j ∂ } , [ x ∂ , X ] = X i + j = ℓ ( x ) i ( x ) j { jc i,j ∂ + ( j − c i,j ∂ } . We add these two expressions to see Θ( X ) = ( ℓ − X . (cid:3) The following result is an analogue of Lemma 3.3; Assertion 2a (resp. Asser-tion 2b) will give rise to Assertion 1 (resp. Assertion 2) of Theorem 3.11.
Lemma 3.14.
Let
M ∈ F B . Suppose that dim { K ( M ) } > . (1) If X ∈ K ( M ) , then X is polynomial in x , i.e. X = n X k =0 ( x ) k X k ( x ) . (2) Choose n = n ( X ) minimal so X ∈ K ( M ) − K B has the form of Assertion 1.Then one of the following two possibilities holds: (a) n = 0 and X = a ( x ) ∂ + a ( x ) ∂ . (b) n = 2 . By making a change of coordinates ( x , x ) → ( x , αx + βx ) ,we can ensure X = 2 x x ∂ + { ( x ) + σ · ( x ) } ∂ for σ ∈ {− , , } .Proof. We use the structure of K ( M ) as a K B module. Let X ∈ K ( M ). Sincead( ∂ ) = ∂ is an endomorphism of K ( M ) and since K ( M ) is finite dimensional,there is a minimal dependence relation of the form: ∂ s X + c s − ∂ s − X + · · · + c X = 0 with s > c i ∈ R . We factor this relation over C to construct a relation w Y v =1 ( ∂ − λ v ) ν v X = 0 for λ v ∈ C distinct and ν v ≥ . We clear the previous notation and let L (resp. L B ) be the complexification of K ( M ) (resp. K B ). Suppose some λ v = 0. By reordering the roots, we may assume λ = 0. Since we have chosen a minimal dependence relation, we have0 = Y := ( ∂ − λ ) ν − w Y v =2 ( ∂ − λ v ) ν v X ∈ L . Since ( ∂ − λ ) Y = 0, Y = e λ x Y ( x ) ∈ L . Since L is finite dimensional, we maychoose Z ∈ L for n maximal of the form0 = Z = e λ x n X k =0 ( x ) k Z k ( x ) for Z n ( x ) not identically zero . We then have 0 = Θ( Z ) = ( x ) n +1 λ e λ x Z n ( x ) + O (( x ) n ) ∈ L which contradictsthe assumption that n was maximal. Thus terms which are true exponentials in x do not occur and the minimal relation for X takes the form ( ∂ ) n X = 0. Thisimplies that X is polynomial in x and establishes Assertion 1.We now establish Assertion 2. Choose X ∈ K ( M ) − K B so that n = n ( X ) isminimal. If n = 0, then X = X ( x ) and Assertion 2a holds. We suppose thereforethat n >
0. One has 0 = ( ∂ ) n X = n ! X n ( x ) ∈ K ( M ). Because n was minimal, X n ( x ) ∈ K B . Since X n ( x ) does not depend on x , X n ( x ) is a constant multipleof ∂ . Therefore after rescaling X if necessary, we may assume X = ( x ) n ∂ + n − X k =0 ( x ) k X k ( x ) ∈ K ( M ) − K B . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 21 If n >
2, then ∂ X ∈ K ( M ) has degree at least 2 in x so ∂ X / ∈ K B . Thiscontradicts the minimality of n and shows n = 1 or n = 2. If n = 1, then˜ X ( x ) := X − x ∂ − x ∂ ∈ K ( M ) − K B is independent of x and thereforehas n ( ˜ X ) = 0; this contradicts the minimality of n . Thus n = 2 so X = ( x ) ∂ + x X ( x ) + X ( x ) ∈ K ( M ) − K B . We note that˜ X ( x ) := ∂ X − x ∂ + x ∂ ) = X ( x ) − x ∂ ∈ K ( M ) . Thus by the minimality of n , Y := X ( x ) − x ∂ ∈ K B . Since Y = Y ( x ), Y = c ∂ . By replacing X by X − c ( x ∂ + x ∂ ), we may assume X = 2 x ∂ so X = ( x ) ∂ + 2 x x ∂ + X ( x ) ∈ K ( M ) − K B . Since ( x ) ∂ + 2 x x ∂ is homogeneous of degree 2, Lemma 3.13 implies Y := (Θ − X = ( x ∂ − X ( x ) ∈ K ( M ) . The minimality of n then shows Y ∈ K B so ( x ∂ − X ( x ) = ǫ∂ . By subtractingan appropriate multiple of ∂ from X we can assume ǫ = 0 so( x ∂ − X ( x ) = 0 . We can solve this ODE to see X is homogeneous of degree 2 in x and, consequently, X = { x x + ( x ) c } ∂ + { ( x ) + ( x ) c } ∂ for some ( c , c ) ∈ R . We consider the linear change of coordinates ( u , u ) = ( x , ǫx + x ). Then ∂ u = ∂ x − ǫ∂ x , ∂ u = ∂ x ,X = { x x + ( x ) c } ∂ x + { ( x ) + ( x ) c } ∂ x = { u ( u − ǫu ) + ( u ) c } ( ∂ u + ǫ∂ u ) + { ( u − ǫu ) + ( u ) c } ∂ u = { u u + ( c − ǫ )( u ) } ∂ u + { ( u ) + ( u ) c ( ǫ ) } ∂ u for c ( ǫ ) ∈ R . Thus by choosing ǫ = c , we may assume c = 0 to assume X = 2 x x ∂ + { ( x ) + ( x ) c } ∂ . We may now rescale x to ensure c ∈ {− , , } . (cid:3) We continue our study firstly assuming the existence of affine Killing vector fieldsas in Assertion 2a of Lemma 3.14. The condition C = C = C = 0 willplay a crucial role in our analysis; by Lemma 2.10, it is an affine invariant in thissetting. The surfaces of Assertion 1 of Theorem 3.11 will arise as follows: Lemma 3.15.
Let
M ∈ F B . (1) x ∂ / ∈ L ( M ) and ( x ) a ∂ ∈ L ( M ) iff a = 1 , C = 0 , C = 0 , C = 0 ,and C = 0 . (2) ( x ) a ∂ ∈ L ( M ) − L B iff a := 1 + C − C = 0 , C = 0 , C = 0 , C = 0 , (3) log( x ) ∂ ∈ L ( M ) iff C − C = 0 , C = 0 , C = 0 , C = 0 . (4) If there exists X = X ( x ) ∈ K ( M ) − K B , then C = C = C = 0 . (5) If C = C = C = 0 , then M is also of Type A , dim { K ( M ) } = 4 , ρ = ( x ) − ˜ ρ dx ⊗ dx , and one of the following holds: (a) C − C = 0 , C = 1 , ˜ ρ = (1 + C ) C = 0 , and K ( M ) = Span { x ∂ − x log( x ) ∂ , x ∂ } ⊕ K B . (b) C = 0 , a = 0 , ˜ ρ = (1 + C − C ) C = 0 , and K ( M ) = Span { x ∂ , ( x ) a ∂ } ⊕ K B . (c) C = 0 , a = 0 , ˜ ρ = ( C ) = 0 , and K ( M ) = Span { x ∂ , log( x ) ∂ } ⊕ K B . Proof.
The first three assertions follow by direct computation. We prove Assertion 4as follows. Suppose that X = a ( x ) ∂ + a ( x ) ∂ ∈ K ( M ) − K B . Let Θ be as definedin Equation (3.e). Because Θ X = ( x ∂ − X , x ∂ X ∈ K ( M ). We factor theminimal dependence relation( x ∂ ) n X + c n − ( x ∂ ) n − X + · · · + c X = 0 for n ≥ C to express this relation in the form s Y v =1 ( x ∂ − λ ν ) ν v X = 0 . Suppose some λ ν = 0. By renumbering the roots, we may suppose a := λ = 0 so0 = Y = ( x ∂ − a ) ν − Y v =2 ( x ∂ − λ ν ) ν v X ∈ L satisfies the ODE ( x ∂ − a ) Y = 0 and hence Y = ( x ) a ( c ∂ + c ∂ ) ∈ L − L . If c = 0, by making a (possibly) complex change of coordinates which takes the form( x , x ) → ( x , x + ǫx ), we may assume Y = ( x ) a ∂ . Assertion 1 then implies a = 1. Therefore, we may take Y to be real and the change of coordinates involvedis real. The relations of Assertion 4 then follow from Assertion 1. If, on the otherhand, c = 0, then Y = ( x ) a ∂ and we use Assertion 2 to show Assertion 4 holds.We may therefore assume the minimal relation takes the form ( x ∂ ) n X = 0 andwe do not need to complexity. If n = 1, then X is constant. Since X ∈ K − K , wemay assume X = ∂ ; this is ruled out by Assertion 1. We therefore conclude that n >
1. By replacing X by ( x ∂ ) n − X if necessary, we may assume without loss ofgenerality that n = 2. Since0 = ( x ∂ ) X ∈ ker { x ∂ } = Span { ∂ , ∂ } ∩ K ( M ) , again by making a change of coordinates of the form ( x , x ) → ( x , x + ǫx ) ifnecessary, we may assume x ∂ X = ∂ . We solve this ODE to see X = log( x ) ∂ + X for X = c ∂ + c ∂ a constant vector field . If c = 1, we renormalize the coordinates so X = ∂ and obtain Killing equations:2 C − C = 0 , C − C − C − ,C − C = 0 , C − C − C = 0 ,C = 0 , C + C = 0 . This implies C = 0, C = − C = 0, C = 0, C = 0, C = 0, and ρ = 0. This is impossible. We therefore have X = log( x ) ∂ and the relations ofAssertion 4 follow from Assertion 3.We impose the relations C = C = C = 0 for the remainder of theproof. If M is also of Type A , then Theorem 3.4 implies dim { K ( M ) } = 4. Thusour task is to construct two additional affine Killing vector fields ξ and ξ so { ξ , ξ , x ∂ + x ∂ , ∂ } are linearly independent and so we can apply Lemma 3.2.If C = 0, then x ∂ ∈ K ( M ) by Assertion 1. We apply Lemma 3.2 to the pair { x ∂ , ∂ } to see M is also of Type A and hence by Theorem 3.4, dim { K ( M ) } = 4.We set a = 1 + C − C . We apply Assertion 2 of Lemma 3.15 to obtainAssertion 5b if a = 0 and to obtain Assertion 5c if a = 0.If C − C = 0, Assertion 2 of Lemma 2.8 shows there is a linear changeof coordinates, which does not affect the normalization C = C = C = 0,to ensure C = 0. The analysis of the previous paragraph then pertains. Wemay therefore assume C − C = 0 and C = 0. By rescaling x , we mayassume C = 1. We apply Assertion 2 with a = 1 to see x ∂ ∈ K ( M ). Adirect computation shows x ∂ − x log( x ) ∂ ∈ M . We apply Lemma 3.2 to the OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 23 pair { x ∂ − x log( x ) ∂ , ∂ } to see M is of Type A and hence by Theorem 3.4dim { K ( M ) } = 4. We then obtain Assertion 5a. (cid:3) Remark 3.16. (1) Observe that C = C = C = 0 in Assertion 4 of Lemma 3.15 is anequivalent condition for a Type B surface to be also of Type A (comparewith the results in [5]).(2) We apply Lemma 3.9 to the three classes in Assertion 5 of Lemma 3.15.Let C define such a connection. Then C transforms to a new connection˜ C = T ∗ b,c C for ˜ C = c ( C + b (2 C − C )) and ˜ C ij k = C ij k otherwise.It now follows that the three classes in Assertion 5 of Lemma 3.15 arelinearly inequivalent surfaces since it is not possible to transform one classinto another using a transformation T b,c .(3) The α invariant satisfies α ( M ) ∈ ( −∞ , ∪ (16 , ∞ ) if M corresponds tothe families 5a and 5b. This shows that these families are affine isomorphicto M c , whereas α ( M ) = 16 for any surface in 5c, and thus they are affineisomorphic to M .Next we assume Assertion 2b of Lemma 3.14 holds. This will give rise to Asser-tion 2 of Theorem 3.11. Lemma 3.17.
Let
M ∈ F B . Assume there exists X ∈ K ( M ) of the form X ( σ ) := 2 x x ∂ + { ( x ) + σ · ( x ) } ∂ for σ ∈ {− , , } . (1) Up to linear equivalence, one of the following possibilities holds: (a) σ = 0 , M = N ± , M is not Type C , and ρ = ± ( x ) − dx ⊗ dx (b) σ = 0 , M = N c , and M is not Type C . (c) σ = 1 , M = M , and M is Type C . (d) σ = − , M = N , and M is Type C . (2) K ( M ) = Span { X ( σ ) } ⊕ K B ≈ su ( , ) . (3) dim { K ( M ) } = 3 , and M is not of Type A . (4) Two different affine surfaces in Definition 3.10 are not locally affine isomor-phic. In particular, linearly equivalent and affine isomorphic are equivalentnotions in this setting.Proof.
Suppose first σ = 0. The Killing equations are C = 0 , C − C + 1 = 0 , C − C = 0 , C + 1 = 0 . We solve these equations to see that C = − , C = 0 , C = − , C = 2 C . If C = 0, we may rescale x to ensure that C = ∓ and obtain the surfaces M ± and compute that ρ = ± ( x ) − dx ⊗ dx so ρ a = 0. The nature of the Riccitensor (see Equation (3.d)) distinguishes these two surfaces.On the other hand, if C = 0, we may rescale x to assume C = 1 and obtainthe surfaces N c . We then have ρ a = ( x ) − dx ∧ dx is invariantly defined.In particular, none of these surfaces is locally isomorphic to N ± . If we express ∇ ρ a = ω ⊗ ρ a , then ω is invariantly defined. We compute ∇ ∂ ρ a = ( x ) − {− − C − C } dx ∧ dx = 0 , ∇ ∂ ρ a = ( x ) − {− C − C } dx ∧ dx = − ( x ) − dx ∧ dx . This shows ˜ ω := ( x ) − dx is invariantly defined. Thus by expressing ρ s = { − c } ( x ) − dx ⊗ dx = { − c } ˜ ω ⊗ ˜ ω , we conclude 1 − c is an affine invariant and hence all these examples are distinctas well.Suppose next σ = 1. The Killing equations are2 C − C = 0 , C − C + 1 = 0 ,C − C + C + 1 = 0 , C − C + C = 0 , C − C = 0 , C − C + 1 = 0 . We solve these relations to see M = N . The symmetric Ricci tensor distinguishesthis surface from the surfaces N ± or N c .Suppose finally σ = −
1. The Killing equations are C + 2 C = 0 , C − C − ,C − C − C + 1 = 0 , C + C − C = 0 , C − C = 0 , C + C + 1 = 0 . We solve these equations to see M = N . The Ricci tensor distinguishes thesesurfaces from the previous examples.We now examine the Lie algebra structure. Let e := X ( σ ), e := − x ∂ − x ∂ , e := − ∂ . We then have [ e , e ] = e , [ e , e ] = e , and [ e , e ] = − e . Theseare the structure equations for su (1 ,
1) given in Equation (3.c). The range of theadjoint map is 3-dimensional; the range of the adjoint map in either A ⊕ A or A , is 2-dimensional. Thus A , = su ( , ) is not a Lie sub-algebra of either A ⊕ A orof A , and hence by Theorem 3.8, M is not of Type A .Let S := X ( σ ) · R ⊕ K B . Suppose to the contrary, there is some additional affineKilling vector field Y ∈ K ( M ) − S . Since M is not of Type A , by Lemma 3.15, Y = Y ( x ). The argument given to prove Lemma 3.14 shows, therefore, ∂ n Y = 0for n ≥
2. If n = 2, then we have that Y = 2 x x ∂ + ( x ) ∂ + Z ( x ) andhence X ( σ ) − Y = σ · ( x ) ∂ − Z ( x ) only depends of x which contradicts theobservation made above. If n >
3, we may replace Y by ( ∂ ) n − Y to ensure n = 3.Since ( ∂ ) ( ∂ Y ) = 0, we conclude ∂ Y must be a multiple of X and hence Y = x ( x ) ∂ + ( ( x ) + σ · ( x ) x ) ∂ + Y ( x ) . We apply Θ − − Y ∈ K B and hence Y = x ( x ) ∂ + ( ( x ) + σ · ( x ) x ) ∂ + ( x ) ( a ∂ + a ∂ ) . We have Killing equations: σ = 0: a + 2 a C = 0, a = 0, 2 − a C = 0. σ = 1: 4 a = 0, 3 a = 0, 2( a −
1) = 0. σ = −
1: 4 a = 0, 3 a = 0, 2(1 + a ) = 0.These equations are inconsistent and thus there is no additional affine Killing vectorfield.Up to linear equivalence and homothety, the only pseudo-Riemannian metricswhich are of Type C have the form ds = ( x ) − (( dx ) + ǫ ( dx ) ). We use theKoszul formula Γ ijk = { g ik/j + g jk/i − g ij/k } to see:Γ = g / = − ( x ) − , Γ = − ( x ) − , Γ = g / = − ( x ) − ǫ, Γ = − ( x ) − , Γ = − g / = ( x ) − ǫ, Γ = ( x ) − ǫ. Taking ǫ = 1 (resp. ǫ = −
1) yields the surfaces N or N . Thus these are of Type C .On the other hand, the symmetric Ricci tensor has rank at most 1 if M = N ± or M = N c so these surfaces are not of Type C . (cid:3) OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 25
Remark 3.18.
In Definition 3.10, we let M := N c for c = be determined by C = − , C = 0 , C = 1 , C = − , C = , C = 2 . The Ricci tensor of this Type B surface is alternating and this affine surface cor-responds to the distinguished situation in [11, Theorem 2-(A.1)]. We shall seepresently that, up to affine equivalence, this is the only affine surface of Type B with dim { K ( M ) } = 3 which admits an affine gradient Ricci soliton.Lemma 2.2 shows that every Type A surface has ρ and ∇ ρ symmetric. If, more-over, Rank { ρ } = 1, then ρ is recurrent (see [5]). The next result shows that thesegeometric conditions identify Type A among Type B surfaces. It is an immediateconsequence of Lemma 2.10 and the discussion of this section. Corollary 3.19.
Let M be a Type B surface which is not flat. The followingconditions are equivalent: (1) M is also of Type A . (2) ρ is symmetric, recurrent, and of rank and ∇ ρ is symmetric. (3) dim { K ( M ) } = 4 . (4) C = C = C = 0 . Remark 3.20.
Note that the geometric conditions given in Corollary 3.19, i.e. ρ is symmetric, recurrent, and of rank 1 and ∇ ρ is symmetric, characterize Type A surfaces amongst Type B ones, but not the converse. In Theorem 3.8, we haveidentified which surfaces of Type A are also of Type B in terms of the α invariantgiven in Definition 2.4 (see Table 1).3.3. Change of coordinates.
The following result is closely related to the workof [9, 15] and deals with the homogeneous affine surfaces where dim { K ( M ) } = 2.As noted in the proof of Lemma 3.9, the Lie group G := { T : ( x , x ) → ( tx , ux + vx + w ) for t > v = 0 } is the 4-dimensional subgroup of GL(2 , R ) which preserves R + × R . Theorem 3.21.
Let M be a simply connected locally homogeneous affine surfacewith dim { K ( M ) } = 2 . (1) If M is Type A , then the coordinate transformations of any Type A atlasfor M take the form ~x → A~x + ~b for A ∈ GL(2 , R ) and ~b ∈ R . (2) If M is Type B , then the coordinate transformations of any Type B atlasfor M belong to G .Proof. Suppose first that M is of Type A . Cover M by Type A coordinate charts( O α , φ α ) so α Γ ∈ R is constant. The transition functions φ αβ then are local dif-feomorphisms of R so that φ ∗ αβ { β ρ } = α ρ . Since dim { K ( M ) } = 2, Theorem 3.4shows that Rank { ρ α } = 2 so α ρ and β ρ define flat pseudo-Riemannian metrics with φ ∗ αβ { β ρ } = α ρ . This implies dφ αβ is constant and, consequently φ αβ is an affinetransformation as given in Assertion 1.Next suppose M is of Type B . Cover M by Type B coordinate charts ( O α , φ α )with transition functions φ αβ . Fix α and β and let ~x = ( x β , x β ) , ~u = ( u α , u α ) , φ αβ = ( x ( u , u ) , x ( u , u )) . We have ∂ u = ∂ u x · ∂ x + ∂ u x · ∂ x and ∂ u = ∂ u x · ∂ x + ∂ u x · ∂ x . Since [ x ∂ x + x ∂ x , ∂ x ] = − ∂ x and dim { K ( M ) } = 2, ∂ x (and similarly ∂ u ) spanthe range of the adjoint action. Consequently ∂ u is a constant multiple of ∂ x . This implies that ∂ u x = 0 and ∂ u x ∈ R so φ αβ = ( x ( u ) , ˜ x ( u ) + cu ) for c ∈ R . We now have that ∂ u = ∂ u x · ∂ x + ∂ u ˜ x · ∂ x , ∂ u = c∂ x , and u ∂ u + u ∂ u = u ∂ u x ∂ x + ⋆∂ x = ǫ ( x ∂ x + x ∂ x ) + ⋆∂ x . This tells us that x = au for some a ∈ R and that ǫ = 1. Consequently φ αβ = ( au , ˜ x ( x ) + cu ) . Therefore ∂ u = a∂ x + ∂ u ˜ x ( x ) ∂ x and ∂ u = c∂ x so u ∂ u + u ∂ u = x ∂ x + x ∂ x + { u ∂ u ˜ x ( u ) − ˜ x ( u ) } ∂ x . Since u ∂ u + u ∂ u ∈ Span { x ∂ x + x ∂ x , ∂ x } , we conclude u ∂ u ˜ x ( u ) − ˜ x ( u ) = d ∈ R . Thus ˜ x ( u ) = bu + d and the coordinate transformation has the desired form. (cid:3) Remark 3.22.
If dim { K ( M ) } >
2, then Theorem 3.11 shows that there are Killingvector fields which do not belong to the Lie algebra of GL(2 , R ). Consequently,there are admissible coordinate transformations which are non-linear. This showsthe condition dim { K ( M ) } = 2 is essential in Theorem 3.21.Lemma 3.6 gives representatives of all the elements in F A which are also ofType B . We now give an explicit identification of those surfaces with elements of F B . In the proof of Theorem 3.8, we showed that every Type A surface whichis also Type B admits coordinates ( x , x ) such that the corresponding Christoffelsymbols Γ kij ∈ R satisfy Γ = Γ = Γ = Γ = 0. We now give an explicitconstruction to show that such elements of F A , which are affine isomorphic to M c or M , are also of Type B . Lemma 3.23.
Let
M ∈ F A satisfy Γ = Γ = Γ = Γ = 0 . We considerthe change of coordinates ( u , u ) = ( e x , x ) . We then have: u Γ = u ( − x Γ ) , u Γ = u x Γ , u Γ ijk = 0 otherwise . Proof.
We compute: du = e x dx , du = dx , ∂ u = e − x ∂ x , ∂ u = ∂ x , ∇ ∂ u ∂ u = e − x {− ∂ x + x Γ ∂ x + x Γ ∂ x } = u ( − x Γ ) ∂ u , ∇ ∂ u ∂ u = e − x { x Γ ∂ x + x Γ ∂ x } = u x Γ ∂ u , ∇ ∂ u ∂ u = x Γ ∂ x + x Γ ∂ x = 0, u Γ = u ( − x Γ ) , u Γ = 0 , u Γ = 0 , u Γ = u Γ , u Γ = 0 , u Γ = 0 . (cid:3) Affine Gradient Ricci Solitons
In this section we study affine gradient Ricci solitons and affine gradient Yamabesolitons. Recall from Definition 1.5 that ( M, ∇ , f ) is an affine gradient Ricci (resp.Yamabe) soliton if H ∇ f + ρ s = 0 (resp. H ∇ f = 0). A ( M ) (resp. Y ( M )) is the spaceof functions on M so that ( M, ∇ , f ) is an affine gradient Ricci (resp. Yamabe)soliton. The following result relates these two notions. Lemma 4.1.
Let M = ( M, ∇ ) be an affine surface. (1) If f ∈ A ( M ) and if X ∈ K ( M ) , then X ( f ) ∈ ker( H ∇ ) , i.e. X ( f ) ∈ Y ( M ) . (2) If h ∈ ker( H ∇ ) , then R ij ( dh ) = 0 for ≤ i < j ≤ m . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 27
Proof.
Let f be an affine gradient Ricci soliton and let X be an affine Killingvector field. We have by naturality that (Φ Xt ) ∗ f is again an affine gradient Riccisoliton. Since the difference of two affine gradient Ricci solitons belongs to ker( H ∇ ),(Φ Xt ) ∗ f − f ∈ ker( H ∇ ). Differentiating this relation with respect to t and setting t = 0 yields Assertion 1. Assertion 2 follows from the identity h ; ijk − h ; ikj = { R kj ( dh ) } i . (cid:3) Type A affine gradient Ricci solitons. Let M be a Type A affine surface.The associated Ricci tensor is symmetric and K A := Span R { ∂ , ∂ } ⊂ K ( M ). Thecomponents of the Hessian are given by: H ∇ ( f ) = − Γ f (0 , − Γ f (1 , + f (2 , , H ∇ ( f ) = H ∇ ( f ) = − Γ f (0 , − Γ f (1 , + f (1 , , H ∇ ( f ) = − Γ f (0 , + f (0 , − Γ f (1 , .If Rank { ρ } = 1, we normalize the coordinate system so ρ = ρ dx ⊗ dx = 0. Weexamine this situation in the following result. Lemma 4.2.
Let M = ( R , ∇ ) where Γ ijk ∈ R and ρ = ρ dx ⊗ dx = 0 . Then f ∈ A ( M ) if and only if f ( x , x ) = ξ ( x ) where ξ ′′ − Γ ξ ′ + ρ = 0 .Proof. Let ρ = ρ dx ⊗ dx = 0. We impose the relations of Lemma 2.3 and setΓ = 0 and Γ = 0. Let f ∈ A ( M ). We have soliton equations f (2 , − Γ f (1 , = 0 ,f (1 , − Γ f (1 , = 0 ,ρ − Γ f (1 , − Γ f (0 , + f (0 , = 0 . We use the first equation to break the analysis into two cases.
Case 1. Γ = 0 . Then f = u ( x ) + u ( x ) e Γ x . The second soliton equationyields u ′ ( x ) − Γ u ( x ) = 0. Thus f = u ( x ) + c e Γ x +Γ x . The finalsoliton equation is u ′′ ( x ) − Γ u ′ ( x ) − ρ ( c e Γ x +Γ x −
1) = 0. This impliesthat c = 0 and that u satisfies the ODE given above. Case 2. Γ = 0 . Then f = u ( x ) + u ( x ) x . We consider the second solitonequation u ′ ( x ) − Γ u ( x ) = 0 to see that f = u ( x ) + c e Γ x x . Hence thefinal soliton equation becomes 0 = − c e x Γ x ρ + ... , where we have omittedterms not involving x . Since ρ = 0, we see that c = 0 so f ( x , x ) = ξ ( x ) and f is a gradient Ricci soliton if and only if ξ satisfies the ODE given. This completesthe proof. (cid:3) The next result shows that gradient Ricci solitons given in Lemma 4.2 are theonly ones in Type A surfaces. Theorem 4.3.
Let M = ( R , ∇ ) where Γ ij k ∈ R and ρ = 0 . The followingassertions are equivalent: (1) Rank { ρ } = 1 . (2) A ( M ) is non-empty.Proof. It is clear from Lemma 4.2 that if Rank { ρ } = 1, then there exists f ∈ A ( M ).We now show that Rank { ρ } = 1 if A ( M ) = { } . Suppose to the contrary thatRank { ρ } = 2; we argue for a contradiction. We necessarily have Rank { R } = 2.We apply Lemma 4.1. If h ∈ ker( H ∇ ), then R ( dh ) = 0 and dh = 0. Thus ker( H ∇ )consists of the constants. Suppose f is a non-trivial gradient Ricci soliton. Since ∂ and ∂ are affine Killing vector fields, ∂ f and ∂ f are constant. This implies f ( x , x ) = ax + bx + c is affine. We may make an affine change of coordinatesto assume f ( x , x ) = x . We shall establish the desired contradiction by showing Γ = 0 and Γ = 0 and then applying Lemma 2.3 to see Rank { ρ } = 1. Thesoliton equations for f ( x , x ) = x are given by:0 = Γ (Γ − Γ ) + Γ ( − Γ + Γ − , − − Γ Γ , Γ + Γ (Γ − Γ ) − Γ Γ − Γ . Again, we examine possibilities:
Case 1. Suppose Γ = 0 . We normalize and set Γ = 1. A soliton equationimplies Γ Γ − Γ + 1 = 0. Thus we set Γ = 1 + Γ Γ . This yields asoliton equation Γ − (Γ ) Γ + Γ (Γ − − . We set Γ = (Γ ) Γ − Γ (Γ − Case 2. Suppose Γ = 0 . If Γ = 0, we have the desired contradiction. Thuswe suppose Γ = 0. By renormalizing x , we may assume Γ = 1. We havesoliton equations 1 + Γ − Γ = 0 and Γ = 0. We set Γ = 1 + Γ andΓ = 0. The final soliton equation then becomes 0 = −
1. This provides thedesired contradiction and completes the proof of the theorem. (cid:3)
Type B affine Gradient Ricci Solitons. Our analysis is similar to that ofSection 4.1 which dealt with Type A surfaces. We compute the components of theHessian in this setting: H ∇ ( f ) = − ( x ) − { C f (1 , + C f (0 , − x f (2 , } , H ∇ ( f ) = H ∇ ( f ) = − ( x ) − { C f (1 , + C f (0 , − x f (1 , } , H ∇ ( f ) = − ( x ) − { C f (1 , + C f (0 , − x f (0 , } . Definition 4.4.
Let ( a, c ) = (0 ,
0) and c ≥
0. Let ˜ c ∈ R . Let P ± a,c and Q ˜ c be theaffine surfaces defined by: P ± a,c : C = (cid:0) a + 4 a ∓ c + 2 (cid:1) , C = c, C = 0 ,C = (cid:0) a + 2 a ∓ c (cid:1) , C = ± , C = ± c, Q ˜ c : C = 0 , C = ˜ c, C = 1 ,C = 0 , C = 0 , C = 1 . Remark 4.5.
We show that Q ˜ c is not flat by computing: ρ ( Q ˜ c ) = ( x ) − (cid:18) − (cid:19) . Similarly, since ( a, c ) = (0 , P ± a,c is not flat by computing: ρ ( P ± a,c ) = ( x ) − (cid:18) a ( ( a + 2) ∓ c ) ± c ∓ c ± a (cid:19) . A direct computation shows a log( x ) ∈ A ( P ± a,c ) and 0 ∈ A ( Q c ). We will show inTheorem 4.12 that none of these surfaces is isomorphic to a different surface andthat P +0 , √ ≈ N where N c : C = − , C = 0 , C = 1 ,C = − , C = c, C = 2is as defined in Definition 3.10. This is the only surface with dim { K ( M ) } = 3 and A non-empty. OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 29
As opposed to Type A surfaces, Type B surfaces do not have symmetric Riccitensor in the generic situation. We recall that as well as there is a one to one relationbetween affine gradient Ricci soliton on an affine surface (Σ , D ) and gradient Riccisoliton on the associated Riemannian extension ( T ∗ Σ , g D ) [3], there is also a one toone relation between Einstein (indeed Ricci flat) Riemannian extensions ( T ∗ Σ , g D )and affine surfaces (Σ , D ) with alternating Ricci tensor [8]. The following resultgives a complete characterization of the elements of F B where ρ is alternating andis a slightly different treatment than that in [11]. Lemma 4.6.
Let
M ∈ F B . The following assertions are equivalent. (1) The Ricci tensor is alternating, i.e., ρ ij = − ρ ji for all ≤ i, j ≤ . (2) 0 ∈ A ( M ) . (3) M is isomorphic to P ± ,c for c > or to Q c for arbitrary c . (4) A ( M ) = R consists of the constant functions.Proof. The equivalence of Assertion 1 and Assertion 2 is immediate. A directcomputation shows ρ is alternating and non-trivial if M is isomorphic to P ± ,c for c = 0 or if M is isomorphic to Q c . Conversely, suppose ρ is alternating. Wedistinguish two cases: Case 1. Suppose C = 0 . We apply Lemma 2.8 to normalize the coordinatesystem so C = 0; we then rescale to assume C = ±
1. We set C = c ; bychanging the sign of x , we may assume c ≥
0. We have ρ s = C ∓ c and ρ = ± ( − C − C ). We set C = ± c and C = 1 + C . We have ρ = C ± c . To ensure ρ = 0, we require c = 0 and hence c >
0. Thus weobtain the relations of P ± ,c : C = 1 ∓ c , C = c, C = 0 , C = ∓ c , C = ± , C = ± c . Case 2. Suppose C = 0 . We set ρ := Γ ij j dx i ; this is invariant under theaction of GL(2 , R ). We compute0 = ρ − ρ = ( x ) − ( C + C ) = ( x ) − ρ ( ∂ ) . We can rescale x to ensure C + C = 2. By replacing ∂ by ∂ − ε∂ forsuitably chosen ε , we may assume ρ ( ∂ ) = ( x ) − { C + C } = 0. We set C = 0, C = − C , and C = 2 − C . We obtain ρ = − x ) − { − C + ( C ) } . If C = 2, we obtain ρ s = ( x ) − which is false. Thus C = 1. We then obtain ρ s = ( x ) − C so we set C = 0. Let C = c ; this is a free parameter. Weobtain the structure Q c .We have shown the equivalence of Assertion 1, Assertion 2, and of Assertion 3.If A ( M ) = R , then 0 ∈ A ( M ) and consequently Assertion 2 holds. Conversely,suppose Assertion 3 holds. Again, we distinguish cases: Case 1. Suppose M = P ± ,c . Let f be a gradient Ricci soliton. A soliton equation0 = ± c f (0 , + x f (1 , implies f (0 , = ( x ) ∓ c f ( x ). We integrate to see that f = ( x ) ∓ c f ( x ) + f ( x ) and obtain a soliton equation0 = c f ( x ) + x {∓ cf ′ ( x ) ∓ ( x ) ± c f ′ ( x ) + x f ′′ ( x ) } . We differentiate with respect to x to see 0 = c f ′ ( x )+ x {∓ cf ′′ ( x )+ x f ′′′ ( x ) } . Set x = 0 to see f ′ ( x ) = 0 since c = 0. Consequently f is constant and f = f ( x ).We now obtain a soliton equation 0 = ∓ x f ′ ( x ) so f is constant as desired. (cid:3) The intersection between Type B and Type A surfaces was previously studiedin Section 3.2 (cf. Corollary 3.19). Now we consider the existence of affine gradientRicci solitons in that particular setting. Lemma 4.7.
Let
M ∈ F B . Assume F also is of Type A . Then f ( x , x ) ∈ A ( M ) if and only if f ( x , x ) = ξ ( x ) where ξ satisfies the ODE (1 + C − C ) C − x C ξ ′ + ( x ) ξ ′′ = 0 .Proof. By Lemma 2.10 and Corollary 3.19 we have that C = 0, C = 0, and C = 0. Then setting ε := (1 + C − C ) C the Ricci tensor takes the form: ρ = ( x ) − (cid:18) ε
00 0 (cid:19) . A soliton equation yields f (0 , ( x , x ) = 0 so f = a ( x ) + b ( x ) x . We obtain asoliton equation C b ( x ) = x b ′ ( x ). This implies f ( x , x ) = a ( x )+ cx ( x ) C .We obtain a single soliton equation:0 = ( x ) a ′′ ( x ) − C x a ′ ( x ) + ε − c C ( x ) C +1 − cx ( x ) C ε . Since ε = 0, we conclude c = 0 and f = f ( x ) satisfies the given ODE. (cid:3) The following is a useful technical result. We adopt the notation of Defini-tion 3.10.
Lemma 4.8.
Assume that
M ∈ F B admits a gradient Ricci soliton. At least oneof the following possibilities holds: (1) C = C = C = 0 , i.e. dim { K ( M ) } = 4 and M also is of Type A . (2) f = a log( x ) ∈ A ( M ) .Proof. We will use Lemma 4.1. Suppose that Rank { R } = 2. If h ∈ ker( H ∇ ),then R ( dh ) = 0 so dh = 0 and h ∈ R is a constant. Let f be an affine gradientRicci soliton. Let ξ = ∂ and ξ = x ∂ + x ∂ be affine Killing vector fields. Sinceker( H ∇ ) = R , ξ f = [ ξ , ξ ] f = ξ ( ξ f ) − ξ ( ξ f ) = 0so f = f ( x ). Furthermore ξ ( f ) = x ∂ f = a ∈ R so, if f ∈ A ( M ), then f ( x ) = a log( x ) as desired.We may therefore assume that Rank { R } = 1. We have ( x ) R is constant.Choose ( α, β ) ∈ R − { } so thatker(R ) = Span { αdx + βdx } . If β = 0, then dx spans ker( R ). If β = 0, let (˜ x , ˜ x ) := ( x , αx + βx ) toensure that d ˜ x spans ker( R ). Thus we may suppose that ker( H ∇ ) = Span { dx i } for i = 1 , Case 1. Suppose that ker( R ) = Span { dx } . Thenker( H ∇ ) = { h : h ( x , x ) = h ( x ) } . Let f ∈ A ( M ). Since ∂ f ∈ ker( H ∇ ), ∂ f is a function of x . This shows that f ( x , x ) = u ( x ) + v ( x ) so the problem decouples. Furthermore, since( x ∂ + x ∂ ) f ∈ ker( H ∇ ) , we have that x ∂ u ( x ) is a function only of x and hence x ∂ u ( x ) ∈ R . Thuswe conclude f ( x , x ) = a log( x ) + v ( x ) . A Ricci soliton equation is:0 = ⋆ + x C v ′ ( x ) − ( x ) v ′′ ( x ) OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 31 where ⋆ indicates a coefficient which is independent of x and of x . From this wesee that v ′′ ( x ) = 0 so v is linear in x and f = a log( x ) + bx . We have a + bx = ( x ∂ + x ∂ ) f ∈ ker( H ∇ ) . Subtracting this from f yields a log( x ) − a ∈ A ( M ) and hence a log( x ) ∈ A ( M )as desired. Case 2. Suppose ker( R ) = Span { dx } . Thus if f ∈ A ( M ), then ∂ f must bea function only of x . Thus f ( x , x ) = u ( x ) + v ( x ) x . Since ( x ∂ + x ∂ ) f also is only a function of x , we obtain the equation ( x v ′ ( x ) + v ( x )) = 0 so v ( x ) = b · ( x ) − and f = u ( x ) + b · ( x ) − x for b ∈ R . There are two subcasesto be considered. Case 2a. Suppose f = u ( x )+ b · ( x ) − x for b = 0 . We normalize x to assume b = 1 so f = u ( x ) + ( x ) − x . We obtain soliton equations:0 = (2 + C ) x + ⋆ ( x ) , C x + ⋆ ( x ) , C x + ⋆ ( x ) . This implies that C = − C = 0 and C = 0. A soliton equation then alsoyields C = 0. This is covered by Assertion 1. Case 2b. Suppose f ( x , x ) = f ( x ) . Assume also f ( x ) = a log( x ) + b so x f ′ ( x ) / ∈ R . We obtain soliton equations:0 = ⋆ + 2 x C u ′ ( x ) , and 0 = ⋆ + x C u ′ ( x ) . We may then conclude that C = 0 and C = 0. A remaining soliton equationthen yields C = 0 which is the case treated in Assertion 1 (cid:3) We can now establish the following classification result.
Theorem 4.9.
Let
M ∈ F B . The space A ( M ) is non-empty if and only if at leastone of the following possibilities holds up to linear equivalence: (1) M is also Type A , i.e., C = 0 , C = 0 , and C = 0 . (2) M is isomorphic to P ± a,c for ( a, c ) = (0 , or to Q c for arbitrary c .Proof. We examine cases. We apply Lemma 4.8. The case C = 0, C = 0, and C = 0 corresponding to Assertion 1 was examined in Lemma 4.7. We completethe proof of Theorem 4.9, by assuming that a log( x ) ∈ A ( M ). If a = 0, then0 ∈ A ( M ). This is the setting of Lemma 4.6; ρ is alternating and we obtain theexamples P ,c for c = 0 or Q c for arbitrary c . We therefore assume a = 0. Wedecompose the analysis into two cases depending on whether C = 0 or C = 0.Suppose first that C = 0. We apply Lemma 2.8 to assume in addition that C = 0. We then obtain three soliton equations:0 = − a (1 + C ) + C + C C − ( C ) + C C , − C C + C , C ( a − C + C + 1) . The second equation implies C = 2 C C and, since C = 0, the thirdequation shows that C = a + C + 1. We rescale x to assume C = ± P ± a,c where c := C .Next suppose that C = 0. We obtain soliton equations0 = C ( − a + 2 C −
1) + C , C ( C − C ) . If C = 0, then we also obtain C = 0. This is the case of Assertion 1 ofTheorem 4.9. We therefore assume C = 0 and obtain C = C . The solitonequation then implies a = C . A final soliton equation then yields C = 0. Thisimplies a = 0 contrary to our assumption. (cid:3) Theorem 4.9 classifies the geometries of Type B surfaces which can admit afunction resulting in an affine gradient Ricci soliton. Generically, either A ( M ) isempty or it is an affine line. For example, Lemma 4.6 shows that surfaces that areisomorphic to P ± ,c for c > Q c for arbitrary c only admit constant functionsas solutions of the Ricci soliton equation (see Statement 2 of Definition 1.5). Thenext theorem shows that A ( M ) is also an affine line for all P ± a,c with a = 0 exceptin two cases: a = − a = − . Theorem 4.10.
Let
M ∈ F B such that the space A ( M ) is neither empty nor anaffine line. Then, up to linear equivalence, one of the following alternatives holds: (1) M is also Type A , i.e., C = 0 , C = 0 , and C = 0 . (2) M = P ±− , and A ( M ) = {− x ) + c x + c } for c i ∈ R . (3) M = P −− ,c , and A ( M ) = {− log( x ) + c ( x − cx ) + c } for c , c ∈ R and c = .Proof. We examine the Hessian. The setting of Assertion 1 in Theorem 4.10 wasexamined previously in Lemma 4.7. We assume the setting of Assertion 2 of The-orem 4.9, i.e. that for a = 0 and c ≥
0, we have: C = (cid:0) a + 4 a ∓ c + 2 (cid:1) , C = c, C = 0 ,C = (cid:0) a + 2 a ∓ c (cid:1) , C = ± , C = ± c, ,ρ = ( x ) − (cid:18) a (4 + 4 a + a ∓ c ) ± c ∓ c ± a (cid:19) . We examine the kernel of the Hessian to determine the most general solution. Let h ∈ ker( H ∇ ) with dh = 0. If h = h ( x ), then H ∇ h = ( x ) − (cid:18) ⋆ ( x ) 00 ∓ h ′ ( x ) (cid:19) . This is not possible since h ′ = 0. Thus h exhibits non-trivial x dependence. Wereturn to the general setting to obtain a relation. To simplify the notation, we leave C as a parameter and obtain:0 = x h (1 , − C h (0 , . This implies h ( x , x ) = ( x ) C u ( x ) + v ( x ). We obtain:0 = ± x v ′ ( x ) − ( x ) C +2 u ′′ ( x ) ± C ( x ) C +1 u ′ ( x ) ± C u ( x )( x ) C . The powers of x decouple. Because h ( x , x ) exhibits non-trivial x dependence,we may conclude that C = 0 and hence C = a + 1. We also conclude u ′′ ( x )must be constant. Let h ( x , x ) = c · ( x ) + c x + v ( x ). We obtain:0 = ∓ c c + 2 c ( x ∓ cx ) ∓ v ′ ( x ) . This ODE implies v is quadratic in x so h ( x , x ) = b · ( x ) + b x + c · ( x ) + c x .We obtain an equation b + ab + cc + 2 ab x + 2 cc x = 0. Since a = 0, b = 0 so h = c ( x ) + c x + b x . We obtain 2 c x ∓ b ∓ cc ∓ cc x = 0. This implies c = 0 so h = c x + b x . The remaining equations become b (1 + a ) + cc = 0 and b + 2 cc = 0 . OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 33
Thus b = − cc . We set c = 1 to take h = x − cx . This yields the finalequation c (2 a + 1) = 0. We require C = ( a + 2 a ∓ c ) = 0. We consider cases: Case 1. Suppose c = 0 . Since C = 0, a + 2 a = 0 so since a = 0, we obtain a = −
2. This yields the possibility of Assertion 2.
Case 2. Suppose a = − . Since C = 0, − ∓ c = 0. Thus C = − c = . This yields the possibility of Assertion 3.This completes the proof ofTheorem 4.10. (cid:3) Remark 4.11.
The proof of Theorem 4.10 is based on the study of the kernel ofthe Hessian on those surfaces that admit an affine gradient Ricci soliton. Thus,the given families result in examples of non-trivial Yamabe solitons of Type B , i.e.with nonconstant potential function:(1) If C = 0, C = 0, and C = 0, then Y ( N ) consists of the solutions f = f ( x , x ) = ξ ( x ) to the ODE − x C ξ ′ + ( x ) ξ ′′ = 0.(2) If M = P ±− ,c for c = 0 or M = P −− ,c for c = , then Y ( M ) consists ofthe functions c ( x − cx ) + c for c , c ∈ R } .4.3. The moduli space of homogeneous affine gradient Ricci solitons.
Thissection is devoted to the proof of the following result, which describes the modulispace of homogeneous affine gradient Ricci solitons.
Theorem 4.12.
Let ( M , ∇ , f ) be a non-flat homogeneous affine gradient Riccisoliton. Then one of the following possibilities holds: (1) ( M , ∇ ) is isomorphic to M ( ∼ = M ∼ = M c for all c ∈ R ), and f ∈ A ( M ) if and only if f ( x , x ) ≡ f ( x ) with f ′′ − f + 2 = 0 . (2) ( M , ∇ ) is isomorphic to M c ( ∼ = M c ) with c (1 + c ) = 0 , and f ∈ A ( M ) ifand only if f ( x , x ) ≡ f ( x ) with f ′′ − (1 + 2 c ) f + c (1 + c ) = 0 . (3) ( M , ∇ ) is isomorphic to M c for all c ∈ [0 , ∞ ) , and f ∈ A ( M ) if and onlyif f ( x , x ) ≡ f ( x ) with f ′′ − cf + (1 + c ) = 0 . (4) ( M , ∇ ) is isomorphic to N ( ∼ = P − ,c for c = √ ), and f ∈ A ( M ) if andonly if f is constant. (5) ( M , ∇ ) is isomorphic to Q c for all c ∈ R , and f ∈ A ( M ) if and only if f is constant. (6) ( M , ∇ ) is isomorphic to P εa,c , where ε = ± , ( a, c ) = (0 , , and (a) if P εa,c = P ε ,c , then f ∈ A ( M ) if and only if f is constant; (b) if P εa,c = P ε − , , then f ∈ A ( M ) if and only if f ( x , x ) = − x ) + c x + c , for c , c ∈ R ; (c) if P εa,c = P −− ,c with c = , then f ∈ A ( M ) if and only if f ( x , x ) = − log( x ) + c ( x − cx ) + c , for c , c ∈ R ; (d) if P εa,c = P ε ,c , P εa,c = P ε − , and P εa,c ( a, c ) = P −− ,c with c = , then f ∈ A ( M ) if and only if f ( x , x ) ≡ f ( x ) = a log( x )+ c , for c ∈ R .The classes listed above represent distinct affine equivalence classes.Proof. Type A affine gradient Ricci solitons are characterized by Theorem 4.3 andLemma 4.2. Thus Assertions 1–3 follow from Lemma 3.6 and Theorem 3.8.Type B affine gradient Ricci solitons are characterized in Theorem 4.9 and The-orem 4.10, thus leading to Assertions 4–6. The only Type B surface with skew-symmetric Ricci tensor and dim { K ( M ) } = 3 is N , which is isomorphic to P − ,c for c = √ . We complete the proof by showing that the affine structures given Assertions1–6 are inequivalent. By Theorem 3.8, classes (1), (2) and (3) have 4-dimensionalKilling algebra A , , A ⊕ A and A , , respectively. Hence these three classes areinequivalent. Adopt the notation of Definition 2.4 to define the α invariant. Wehave α ( M ) = 16, α ( M c ) = ( c + c ) − c ) , and α ( M c ) = (1 + c ) − c .This shows that if i = 3 , M ci ∼ = M ˜ ci if and only if c = ˜ c . By Theorem 3.11, K ( N ) = su (1 ,
1) and hence N is not affine isomorphic to (1), (2) or (3). Classes(5) and (6) are of Type B and have Killing algebra of dimension 2, which showsthat they are not isomorphic to any of the other classes.We now show the surfaces in Assertions 5 and 6 are inequivalent as well. Set: ρ := x { Γ dx ⊗ dx + Γ dx ⊗ dx − Γ dx ⊗ dx − Γ dx ⊗ dx } ,ρ := Γ ijk Γ kll dx i ⊗ dx j , ρ := Γ ikl Γ jlk dx i ⊗ dx j , ρ := Γ ijj dx i . By Theorem 3.21 1, the coordinate transformations of any Type B surface M withdim { K ( M ) } = 2 belong to the Lie group G which is a subgroup of GL(2 , R ). Sincecontracting an upper against a lower index is a GL(2 , R ) invariant, the tensors { ρ , ρ , ρ } are invariantly defined on any such surface. Since we may express ρ = ρ + ρ − ρ , we conclude that ρ is invariantly defined as well; ρ is a G invariant but not a GL(2 , R ) invariant. We note that ρ is skew-symmetric forany surface Q ˜ c and that ρ ( ∂ , ∂ ) = 0 for any surface P εa,c . Hence no surface inAssertion 5 may be equivalent to any surface in Assertion 6.The invariant ρ is a symmetric (0 , Q c which is given by ρ ( Q c ) = 2( x ) − ( c dx ⊗ dx + dx ⊗ dx ) . It defines a pseudo-Riemannian metric of constant curvature − c − if c = 0. Thisshows that Q c ∼ = Q ˜ c if and only if c = ˜ c . We apply Lemma 3.9. The pull-backaction of T b,c rescales ∂ : ( T b,c ) ∗ ∂ = ∂ + b∂ and ( T b,c ) ∗ ∂ = c∂ . The surfaces P εa,c satisfy ρ ( ∂ , ∂ ) = − ǫ ( x ) − and ρ ( ∂ , ∂ ) = 2 cǫ ( x ) − . Consequently, if P εa,c is affine isomorphic to P ˜ ε ˜ a, ˜ c then ǫ = ˜ ǫ and c = ˜ c . We compute that ρ ( ∂ , ∂ ) = ( x ) − ǫ (1 + 3 a + a + 2 c ǫ ) ,ρ ( ∂ , ∂ ) = ( x ) − ǫ (2 a + a + 2 c ǫ ) . This implies that a = ˜ a which completes the proof. (cid:3) Geodesic completeness.
We have the following application of our analysis.
Lemma 4.13.
Let M be a locally homogeneous surface of Type A which is notsymmetric and with Rank { ρ } = 1 . Then M is not geodesically complete.Proof. The analysis of Section 3 shows that in any Type A chart ( x , x ), the affineKilling vector fields are real analytic. If ( u , u ) is another Type A chart whichintersects the given one, then ∂ u and ∂ u are affine Killing vector fields and hencereal analytic. This implies that M is a real analytic surface with respect to an atlasof Type A charts and our analysis shows A ( M ) consists of real analytic functions on M . We suppose Rank { ρ } = 1 and apply Lemma 2.3 to see Γ = 0 and Γ = 0.We have ∇ ρ = − dx ⊗ ρ . Since M is not symmetric, Γ = 0, and we canfurther normalize the coordinates so Γ = 1. Let σ ( t ) := ( x ( t ) , x ( t )) be a localgeodesic. The geodesic equations become ¨ x ( t ) + ˙ x ( t ) ˙ x ( t ) = 0 which may besolved by setting x ( t ) = log( t ) for t ∈ ( t , t ) some appropriate positive interval.By Lemma 4.2, ξ ( x ) = ρ x ∈ A ( M ). Since M is simply connected, we canextend ξ to a global element of A ( M ) which is real analytic. Furthermore, since M is geodesically complete, we can extend σ to a global real analytic geodesic. Since ξ ( σ ( t )) = ρ log( t ) for t ∈ ( t , t ), ξ ( σ ( t )) = ρ log( t ) for all t ∈ R ; this is notpossible. (cid:3) OMOGENEOUS AFFINE GRADIENT RICCI SOLITONS 35
In memory of the attacks in Beirut and Paris in November 2015.Solidarit´e.
References [1] M. Brozos-V´azquez, E. Calvi˜no-Louzao, E. Garc´ıa-R´ıo, and R. V´azquez-Lorenzo, “Local struc-ture of self-dual gradient Yamabe solitons”. Geometry, Algebra and applications: From Me-chanics to Crytography. In Honor of Jaime Mu˜noz Masqu´e, Springer Proc. Math. Stat., toappear.[2] T. Arias-Marco and O. Kowalski, “Classification of locally homogeneous affine connectionswith arbitrary torsion on 2-manifolds”,
Monatsh. Math. (2008), 1–18.[3] M. Brozos-V´azquez and E. Garc´ıa-R´ıo, “Four-dimensional neutral signature self-dual gradientRicci solitons”, arXiv:1410.8654.[4] M. Brozos-V´azquez, E. Garc´ıa-R´ıo, and P. Gilkey, “Homogeneous affine surfaces: Modulispaces” (in preparation).[5] E. Calvi˜no-Louzao, E. Garc´ıa-R´ıo, and R. V´azquez-Lorenzo, “Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor”,
Canad. J. Math. (2010), 1037–1057.[6] A. Derdzinski, “Noncompactnes and maximum mobility of Type III Ricci-flat self-dual neutralWalker four-manifolds”, Q. J. Math. (2011), 363–395.[7] E. Garc´ıa-R´ıo, P. Gilkey, and S. Nikˇcevi´c, “Homothety Curvature Homogeneity and HomothetyHomogeneity”, Ann. Global Anal. Geom. (2015), 149–170.[8] E. Garc´ıa-R´ıo, D. Kupeli, M. E. V´azquez-Abal, R. V´azquez-Lorenzo, “Affine Osserman con-nections and their Riemannian extensions”, Differential Geom. Appl. (1999), 145–153.[9] A. Guillot and A. S´anchez-Godinez, “A classification of locally homogeneous affine connectionson compact surfaces”, Ann. Global Anal. Geom. (2014), 335–349.[10] S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry vol. I and II”, WileyClassics Library . A Wyley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.[11] O. Kowalski, B. Opozda, and Z. Vlasek, “A classification of locally homogeneous affine con-nections with skew-symmetric Ricci tensor on 2-dimensional manifolds”,
Monatsh. Math. (2000), 109–125.[12] O. Kowalski, B. Opozda, and Z. Vlasek, “On locally nonhomogeneous pseudo-Riemannianmanifolds with locally homogeneous Levi-Civita connections”.
Internat. J. Math. (2003),559–572.[13] O. Kowalski, Z. Vlasek, “On the local moduli space of locally homogeneous affine connectionsin plane domains”, Comment. Math. Univ. Carolina (2003), 229–234.[14] B. Opozda, “A classification of locally homogeneous connections on 2-dimensional manifolds”, Differential Geom. Appl. (2004), 173–198.[15] B. Opozda, “Locally homogeneous affine connections on compact surfaces”, Proc. Amer.Math. Soc. (2004), 2713–2721.[16] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Invariants of real low dimensionLie algebras”,
J. Mathematical Phys. (1976), 86–994. MBV: Departmento de Matem´aticas, Escola Polit´ecnica Superior, Universidade daCoru˜na, Spain
E-mail address : [email protected] EGR: Faculty of Mathematics, University of Santiago de Compostela, 15782 Santi-ago de Compostela, Spain
E-mail address : [email protected] PG: Mathematics Department, University of Oregon, Eugene OR 97403, USA
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