Codimension one Ricci soliton subgroups of solvable Iwasawa groups
Miguel Dominguez-Vazquez, Victor Sanmartin-Lopez, Hiroshi Tamaru
aa r X i v : . [ m a t h . DG ] F e b CODIMENSION ONE RICCI SOLITON SUBGROUPSOF SOLVABLE IWASAWA GROUPS
MIGUEL DOM´INGUEZ-V ´AZQUEZ, V´ICTOR SANMART´IN-L ´OPEZ, AND HIROSHI TAMARU
Abstract.
Recently, Jablonski proved that, to a large extent, a simply connected solv-able Lie group endowed with a left-invariant Ricci soliton metric can be isometricallyembedded into the solvable Iwasawa group of a non-compact symmetric space. Motivatedby this result, we classify codimension one subgroups of the solvable Iwasawa groups ofirreducible symmetric spaces of non-compact type whose induced metrics are Ricci soli-tons. We also obtain the classifications of codimension one Ricci soliton subgroups ofDamek-Ricci spaces and generalized Heisenberg groups. Introduction
The investigation of homogeneous Einstein manifolds and, more recently, homogeneousRicci solitons constitutes an important area of research in differential geometry. By theirvery nature, these geometric structures have typically been studied with Lie theoretic meth-ods in combination with tools of intrinsic Riemannian geometry. In spite of outstandingrecent progress (e.g. [16, 17, 20, 21, 22]), the current understanding is not yet complete,and the classification problem seems to be rather difficult. Several nice classification resultshave been obtained, but mainly in low dimensions (see for example [1, 2, 14, 26]).Recently, Jablonski [18] proved the following remarkable result, which lies at the inter-section of Ado’s and Nash’s embedding theorems: every simply connected Ricci solitonsolvmanifold (in particular, every Einstein solvmanifold) or -step nilpotent Lie group withleft-invariant metric can be realized as a submanifold of a symmetric space . Let us be moreprecise. Let M ∼ = G/K be a symmetric space of non-compact type. The connected realsemisimple Lie group G admits an Iwasawa decomposition G ∼ = KAN . This allows us toidentify, as Riemannian manifolds, M with the solvable Iwasawa group AN endowed withcertain left-invariant metric (see § S be a simply connectedsolvable Lie group with a left-invariant metric (solvmanifold), and assume that S is either Mathematics Subject Classification.
Key words and phrases.
Ricci soliton, solvsoliton, nilsoliton, hypersurface, codimension one, generalizedHeisenberg group, Damek-Ricci space, symmetric space, Iwasawa group.The first two authors have been supported by projects PID2019-105138GB-C21 (AEI/FEDER, Spain)and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain). The first author acknowledges supportof the Ram´on y Cajal program of the Spanish State Research Agency. The third author was supported byJSPS KAKENHI Grant Number JP19K21831. This work was partially supported by the Fonds Weten-schappelijk Onderzoek – Vlaanderen (FWO), the Fonds de la Recherche Scientifique – FNRS under EOSProject No. G0H4518N, and Osaka City University Advanced Mathematical Institute (MEXT Joint Us-age/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). completely solvable and Ricci soliton, or 2-step nilpotent. Then, Jablonski proved thatthere is an n ∈ N and an injective Lie group homomorphism φ : S → AN that is alsoan isometric embedding, where AN is the solvable Iwasawa group of the symmetric space SL ( n, R ) / SO ( n ), endowed with its natural (unique up to scaling) left-invariant Einsteinmetric. Since any Ricci soliton solvmanifold (and in particular, any Einstein solvmanifold)is isometric to a completely solvable Lie group with a left-invariant metric [17], one getsthe result emphasized above.Therefore, these results ensure that some important families of homogeneous Ricci soli-tons always arise as submanifolds of the solvable Iwasawa group associated with a symmet-ric space of non-compact type. This opens up the possibility of addressing the investigationof homogeneous Ricci solitons from the viewpoint of extrinsic submanifold geometry in sym-metric spaces . In this sense, on the one hand one may aim to have some interesting familiesof examples, and on the other hand to investigate general properties of such submanifoldsand, ideally, derive classification results.Regarding the first problem, there are two important collections of examples of Einsteinsubmanifolds of symmetric spaces of non-compact type. One of them is made of theirreducible totally geodesic submanifolds. This is a set of somehow trivial examples froman intrinsic viewpoint, as these submanifolds are themselves symmetric spaces. But one hasto emphasize that the classification problem of totally geodesic submanifolds in symmetricspaces is still outstanding (see for example [5] for a recent contribution). The secondset of examples is due to the third author [25], who proved that the solvable part A Φ N Φ of a parabolic subgroup of the (real semisimple) isometry group of a symmetric spaceof non-compact type M is an Einstein solvmanifold (arising as an equivariant isometricembedding into the corresponding Iwasawa group AN ). Here, Φ is an arbitrary subset ofsimple restricted roots of the symmetric space. These submanifolds are interesting in thatthey provide examples of Einstein solvmanifolds with nilradicals of arbitrary large degreesof nilpotency. Moreover, from an extrinsic point of view, they are minimal (and, except insome limit cases, not totally geodesic) submanifolds.In view of the existence of general embedding results and some interesting families ofexamples, we propose to undertake the project of investigating homogeneous Ricci solitonsfrom the perspective of extrinsic submanifold geometry. In the present article, we willrestrict our attention to submanifolds of codimension one. The reason for this is that, asis standard in submanifold geometry, the codimension one case is the first step towards amore general investigation, and at the same time it usually provides interesting geometricphenomena and examples. However, it is important to mention at this point that we can-not expect to gather in our classifications some of the examples of Einstein submanifoldsmentioned above, except for some trivial cases. Indeed, it is well-known that the only irre-ducible symmetric spaces which admit totally geodesic hypersurfaces are those of constantcurvature and, by definition, none of the examples in [25] is of codimension one either.Our first result is the classification of codimension one subgroups of the solvable Iwasawagroup AN of an irreducible symmetric space of non-compact type M which are Riccisolitons. Theorem A.
Let S be a codimension one connected Lie subgroup of the solvable Iwasawagroup AN of an irreducible symmetric space of non-compact type M . Then S is a Riccisoliton with respect to the metric induced by the left-invariant Einstein metric on AN ifand only if: (i) M is arbitrary and S contains the nilpotent part N , (ii) M is a complex hyperbolic plane C H and S is a Lohnherr hypersurface W , or (iii) M is a real hyperbolic space R H n and S is arbitrary. In item (i), S is a codimension one connected Lie subgroup of AN containing N if andonly if its Lie algebra is of the form ( a ⊖ ℓ ) ⊕ n , where a ⊕ n is the Lie algebra of AN , ⊖ denotes orthogonal complement, and ℓ is a one-dimensional vector subspace of a . In therank one case, such an S is precisely N . In the higher rank case, there are continuouslymany isometry classes of such subgroups S [8], and some of them are Einstein [11, § M , that is, the levelsets of Busemann functions [13, § W n − (also called a fan) is the unique (up to congruence) minimalhomogeneous hypersurface of a complex hyperbolic space C H n [4]. It can be defined as theconnected Lie subgroup of AN ∼ = C H n with Lie algebra a ⊕ ( n ⊖ ℓ ), where ℓ is a one-dimensional subspace of the only simple root space g α ⊂ n = g α ⊕ g α . Finally, in relationto item (iii) of Theorem A, we comment that any codimension one subgroup of the Iwasawagroup of a real hyperbolic space is a space form of constant curvature, and hence Einstein.We will sketch below in this introduction the common approach to the main results ofthis paper, including Theorem A. But at this point we would like to draw the attention tothe main difficulties for proving Theorem A. One is that codimension one subgroups S of AN form a rich class. There is a one-parameter family of isometry classes of such S forthe rank one case, and even a larger family for the higher rank case, depending on the rootsystem. We need to study the Ricci soliton condition for all of them. Another difficulty isthe very determination of the Ricci tensor of the hypersurface S of M . The reason is thatthe Levi-Civita connection of a symmetric space of non-compact type, despite admittinga Lie algebraic description, turns out to be quite hard to handle in full generality, since itinvolves Lie brackets which relate (positive) root spaces in a complicated way. One signof this difficulty is that, previously to this article, a classification as in Theorem A hadonly been obtained, apart from the well-known case of constant curvature M ∼ = R H n , inthe very specific setting of the complex hyperbolic space M ∼ = C H n [15]. Note that bothcases are of rank one, and hence the associated root systems only have one simple root.There are some other studies in particular rank two spaces [24], but the hypersurfacesare assumed to satisfy some additional conditions. In this paper, we address and handlethe problem for a general root system, yielding the (as far as we know) first general andsystematic investigation of Ricci soliton hypersurfaces in the whole family of symmetricspaces of non-compact type.We would also like to note that Theorem A could be stated in a more general wayby assuming that S is a Lie hypersurface of M . Here, by Lie hypersurface we mean acodimension one orbit of a cohomogeneity one action on M with no singular orbits. Lie M. DOM´INGUEZ-V ´AZQUEZ, V. SANMART´IN-L ´OPEZ, AND H. TAMARU hypersurfaces have been classified by Berndt and Tamaru [7], and they turn out to be con-gruent, precisely, to codimension one Lie subgroups of the solvable Iwasawa groups. How-ever, unlike other ambient manifolds (such as Euclidean spaces, spheres or, more generally,irreducible symmetric spaces of compact type), the classification of cohomogeneity one ac-tions (or, equivalently, homogeneous hypersurfaces) on symmetric spaces of non-compacttype is still an outstanding open problem. Although there are partial classification results(see [8]), and in particular Lie hypersurfaces have been classified [7], there is a remarkablerichness of examples, and analyzing their geometry is usually a difficult problem; see [12]for a recent contribution.The proof of Theorem A relies on the study of two different cases: rank one and higherrank. Whereas, as already mentioned, the higher rank cases require a careful analysis ofthe geometry of Lie hypersurfaces in terms of restricted root systems, the case of rankone symmetric spaces is subsumed into the investigation of a broader family of Einsteinsolvmanifolds, namely, Damek-Ricci harmonic spaces [9]. These are well-known solvableextensions of the so-called generalized Heisenberg groups (or H -type groups), which in turnconstitute an important family of two-step nilpotent metric Lie groups. Roughly speaking(see § N with Lie algebra n = v ⊕ z , where z is the center of n , v is a Clifford module over z , and N is endowed with certain natural left-invariant metric. A Damek-Ricci space is a simplyconnected semidirect product AN with left-invariant metric, where N is a generalizedHeisenberg group and A is a one-dimensional Lie group (see § R H n ) are, indeed,the only symmetric Damek-Ricci spaces.In line with our interest in codimension one in this paper, we will first derive the classi-fication of codimension one Ricci soliton subgroups of generalized Heisenberg groups. Wedenote a generalized Heisenberg group by N ( m, k ) or N ( m, k + , k − ), where m is the dimen-sion of the center, and k or ( k + , k − ) represents the number of irreducible factors of thecorresponding Clifford modules; we refer to § Theorem B.
Let S be a codimension one connected Lie subgroup of a generalized Heisen-berg group N . Let s = n ⊖ R ξ be the corresponding Lie subalgebra of n . Then ξ ∈ v , and S is a Ricci soliton with respect to the induced metric if and only if N is isometric to: (i) N (1 , k ) for some k ≥ , (ii) N (2 , , (iii) N ( m, , , where m ∈ { , } , or (iv) N ( m, , where m ∈ { , } , and ξ belongs to one of the two m -dimensional irreduciblehalf-spin submodules of the Spin ( m ) -module v . Some of the Ricci soliton nilmanifolds S obtained above would be interesting also froman intrinsic viewpoint. The most complicated one is the codimension one subgroup S in N (8 , S is a two-step nilpotent 23-dimensional Lie group with 8-dimensionalcenter, which would have not been considered in the literature as far as the authors know.The analysis developed in the proof of Theorem B will then be useful to obtain theclassification in Damek-Ricci spaces. Theorem C.
Let S be a codimension one connected Lie subgroup of a Damek-Ricci space AN . Then S is a Ricci soliton with respect to the induced metric if and only if S = N is the generalized Heisenberg group associated with AN , or AN is isometric to a complexhyperbolic plane C H and S is the Lohnherr hypersurface W . The proofs of Theorems A, B and C rely on the same simple underlying idea: use theGauss equation of submanifold geometry to calculate the Ricci operator of an arbitrarycodimension one subgroup, and then decide when the induced metric of such subgroup isan algebraic Ricci soliton. We recall that a left-invariant metric g on a Lie group S is analgebraic Ricci soliton if there exist a derivation D ∈ Der( s ) and a real number c such thatRic = c Id + D, where Ric is the (1 , S, g ). Algebraic Ricci solitons on simply connectedgroups are Ricci solitons and, indeed, all known examples of expanding homogeneous Riccisolitons are isometric to simply connected solvsolitons; we recall that a solvsoliton (resp.nilsoliton) is, precisely, a solvable (resp. nilpotent) Lie group S with an algebraic Riccisoliton metric. Various fundamental results have been proved over the last few yearsregarding the relation between homogeneous Ricci solitons and algebraic Ricci solitons(see for example [16, 17, 20, 21, 22]). In particular, it is known that any left-invariant Riccisoliton metric on a completely solvable (or in particular, nilpotent) Lie group is necessarilya solvsoliton. Since all codimension one subgroups S studied in this paper are completelysolvable, we can reduce our problem of determining when S is a Ricci soliton (with theinduced left-invariant metric) to the analysis of the algebraic Ricci soliton condition on S .As mentioned above, the first basic step in our arguments is to calculate the Ricci opera-tor of each codimension one subgroup S . This is done via the Gauss equation in terms of theshape operator and the normal Jacobi operator of S . Whereas for generalized Heisenberggroups and Damek-Ricci spaces (Theorems B and C) this is a relatively straightforwardtask, dealing with symmetric spaces (Theorem A) is much more involved, as this requires ameticulous inspection of the extrinsic geometry of S in terms of the root space decomposi-tion of the isometry Lie algebra of the symmetric space. Once we have calculated the Riccioperator Ric of S , the classifications are reduced to the determination of the conditionsfor which the operator Ric − c Id is a derivation of s , for some c ∈ R . This is a purelyalgebraic but non-trivial problem which requires different approaches depending on theambient space where S lives. Thus, for generalized Heisenberg groups and Damek-Riccispaces we develop an analysis involving Clifford modules and spin representations, whereasfor symmetric spaces we are led again to arguments involving root spaces.This article is structured as follows. In Section 2 we basically fix some notation andrecall some well-known facts concerning Riemannian submanifold geometry. Section 3 isdevoted to the study of Ricci soliton hypersurfaces in generalized Heisenberg groups andthe proof of Theorem B. Section 4 deals with Damek-Ricci spaces and contains the proof ofTheorem C. Finally, in Section 5 we investigate Ricci soliton Lie hypersurfaces in symmetricspaces of non-compact type, and derive the proof of Theorem A. Each one of these threemain sections contains a subsection with preliminaries on generalized Heisenberg groups,Damek-Ricci spaces and symmetric spaces of non-compact type, respectively. M. DOM´INGUEZ-V ´AZQUEZ, V. SANMART´IN-L ´OPEZ, AND H. TAMARU
Acknowledgments.
The authors would like to thank Eduardo Garc´ıa-R´ıo and AlbertoRodr´ıguez-V´azquez for some helpful comments.2.
Preliminaries
In this short section we introduce some basic notation and facts concerning Riemanniangeometry of hypersurfaces.Let M be a Riemannian manifold with metric h· , ·i and Levi-Civita connection ∇ . We willadopt the sign convention R M ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z for the definitionof the curvature tensor R M of M , where X , Y , Z are smooth vector fields on M . We willdenote the Ricci operator of M by Ric M .Now, let S be a hypersurface of the Riemannian manifold M . Let ξ be a smooth unitnormal vector field on (an open subset of) S . The shape operator of S is the endomorphism S ξ of T S given by S ξ X := − ( ∇ X ξ ) ⊤ = −∇ X ξ , for each X ∈ T S , where ⊤ denotesorthogonal projection onto the tangent space of S . The shape operator is a self-adjointendomorphism of the tangent space of S . The principal curvatures of S are preciselythe (real) eigenvalues of S ξ , and the mean curvature of S is the trace tr S ξ . Anotherimportant object which relates the geometry of the ambient space M with that of thehypersurface S is the Jacobi operator of S , defined as the endomorphism R ξ of T S givenby R ξ ( X ) := R M ( X, ξ ) ξ , for each X ∈ T S . Again, the Jacobi operator is self-adjoint.The extrinsic geometry of a submanifold is controlled by the fundamental equations ofsubmanifold geometry. One of these relations is Gauss equation, which, for a hypersurface S of M as above, can be written as h R M ( X, Y ) Z, W i = h R ( X, Y ) Z, W i − hS ξ Y, Z ihS ξ X, W i + hS ξ X, Z ihS ξ Y, W i , for tangent vectors X , Y , Z , W to S , and where R is the curvature tensor of S . From thisequation, one can easily derive the following expression for the Ricci operator Ric of S interms of the Ricci operator of M and the shape and Jacobi operators of S :(1) Ric = (Ric M | T S ) ⊤ + tr( S ξ ) S ξ − S ξ − R ξ . In this article we will work with some particular types of homogeneous hypersurfacesin different ambient spaces. Recall that by a homogeneous hypersurface we understand acodimension one orbit S of an isometric action of a (connected) Lie group on an ambientspace M . Homogeneous hypersurfaces have constant principal curvatures with constantmultiplicities, and the same happens with the eigenvalues of the Jacobi operator. In thispaper, M will always be isometric to a Lie group with a left-invariant metric, and S acodimension one Lie subgroup of M . In this setting, the tangent space of S at each pointis spanned by the left-invariant vector fields in the Lie algebra s of S , and hence ξ is aleft-invariant unit normal vector field globally defined on S . Nilsoliton hypersurfaces in generalized Heisenberg groups
In this section we determine which codimension one Lie subgroups of generalized Heisen-berg groups are Ricci solitons with the induced metric. In § § Generalized Heisenberg groups.
Let n = v ⊕ z be a Lie algebra equipped with a positive definite inner product h· , ·i suchthat h v , z i = 0, and whose Lie bracket satisfies [ v , v ] ⊂ z and [ n , z ] = 0. In this section wewill assume that v = 0 = z , since otherwise n would be an abelian Lie algebra. Define alinear map J : z → End( v ) by h J Z U, V i = h [ U, V ] , Z i , for all U , V ∈ v , Z ∈ z .Then, the two-step nilpotent Lie algebra n is called a generalized Heisenberg algebra or H-type algebra if J Z = −h Z, Z i Id , for all Z ∈ z .The simply connected nilpotent Lie group N with Lie algebra n , equipped with the left-invariant metric induced by h· , ·i , is called a generalized Heisenberg group or H-type group .The map J induces a representation of the Clifford algebra Cl ( m ) = Cl ( z , q ) on v , where m = dim z and q = −h· , ·i| z × z is a negative definite quadratic form. Conversely, a repre-sentation of such a Clifford algebra induces a map J as above and, hence, a generalizedHeisenberg group. Therefore, the classification of these metric Lie groups follows fromthe classification of Clifford modules. If m d over Cl ( m ), up to equivalence, and each Clifford module over Cl ( m ) is isomorphic to v ∼ = ⊕ k d . We will denote the corresponding generalized Heisenberggroup by N ( m, k ). If m ≡ d + and d − , over Cl ( m ), up to equivalence; they satisfy dim d + = dim d − , and eachClifford module over Cl ( m ) is isomorphic to v ∼ = (cid:0) ⊕ k + d + (cid:1) ⊕ (cid:0) ⊕ k − d − ). We will denote thecorresponding group by N ( m, k + , k − ). In any case, we will denote by k the number ofirreducible Cl ( m )-submodules in v ; thus, k = k + + k − if m ≡ n = dim v , we have n = k dim d ( ± ) . As vector spaces, the irreducible modules d or d ± areisomorphic to the vector spaces shown in Table 1, where the explicit dimensions of v arealso included. Moreover, if m ≡ N ( m, k + , k − ) is isometric to N ( m, k ′ + , k ′− )if and only if ( k ′ + , k ′− ) ∈ { ( k + , k − ) , ( k − , k + ) } . m p p + 1 8 p + 2 8 p + 3 8 p + 4 8 p + 5 8 p + 6 8 p + 7 d ( ± ) R p C p H p H p H p +1 C p +2 R p +3 R p +3 n p k p +1 k p +2 k p +2 k p +3 k p +3 k p +3 k p +3 k Table 1.
Clifford modules.
M. DOM´INGUEZ-V ´AZQUEZ, V. SANMART´IN-L ´OPEZ, AND H. TAMARU
Let
U, V ∈ v and X, Y ∈ z . Then the following relations hold:(2) h J X U, J X V i = | X | h U, V i , h J X U, J Y U i = h X, Y i| U | , [ U, J X U ] = | U | X. In particular, the first and third equalities imply that J X is an orthogonal complex structureon v for any unit X ∈ z , and [ v , v ] = z . Finally, we recall the formula of the Levi-Civitaconnection of a generalized Heisenberg group,(3) ∇ V + Y ( U + X ) = − J X V − J Y U −
12 [
U, V ] , which will be fundamental in our work, as well as the Ricci operator Ric N of a generalizedHeisenberg group N ,(4) Ric N | v = − m , Ric N | z = n . We refer to [9, Chapter 3] for more details on these facts. We conclude this subsectionby observing that any generalized Heisenberg group is an algebraic Ricci soliton. In fact,it is straightforward that the endomorphism D of n = v ⊕ z determined by D | v = Idand D | z = 2 Id is a derivation of n , and Ric N = e D + c Id with e D = ( n/ m/ D and c = − m − n/ Proof of the classification result.
Let S be a connected Lie subgroup of codimension one of a generalized Heisenberggroup N . Let s = n ⊖ R ξ be the corresponding Lie subalgebra of n = v ⊕ z , where ⊖ denotes (here and henceforth) orthogonal complement, and ξ = W + Z ∈ n is a unit vectorwith W ∈ v , Z ∈ z . Then J Z W , | Z | W − | W | Z ∈ s , and hence by (2)(5) [ J Z W, | Z | W − | W | Z ] = −| Z | | W | Z ∈ s . This, together with [ v , v ] = z , implies that ξ ∈ v . Conversely, if ξ ∈ v , then one can easilysee that s = n ⊖ R ξ = ( v ⊖ R ξ ) ⊕ z is a Lie subalgebra of n .In what follows, we will use the following notation: J = { J Z : Z ∈ z } , J ξ = { J Z ξ : Z ∈ z } , and ( J ξ ) ⊥ = v ⊖ ( R ξ ⊕ J ξ ) . Note that we have the orthogonal direct sum decomposition s = J ξ ⊕ ( J ξ ) ⊥ ⊕ z , wheredim J ξ = dim z = m and dim( J ξ ) ⊥ = n − m − S ξ of the hypersurface S of N by using (3)along with the relations (2): S ξ U = −∇ U ξ = 12 [ ξ, U ] = 0 , for any U ∈ ( J ξ ) ⊥ , S ξ J Z ξ = −∇ J Z ξ ξ = 12 [ ξ, J Z ξ ] = 12 Z, for any Z ∈ z , J Z ξ ∈ J ξ, S ξ Z = −∇ Z ξ = 12 J Z ξ, for any Z ∈ z . Hence, both ( J ξ ) ⊥ and J ξ ⊕ z are invariant under S ξ and, moreover,(6) tr( S ξ ) = 0 , S ξ | ( J ξ ) ⊥ = 0 , S ξ | J ξ ⊕ z = 14 Id . Now, we calculate the normal Jacobi operator R ξ = R N ( · , ξ ) ξ of S by combining thedefinition of the curvature tensor of the ambient space, R N ( X, Y ) = [ ∇ X , ∇ Y ] − ∇ [ X,Y ] ,with the formula (3) for the Levi-Civita connection, or alternatively by using the formulafor the Jacobi operator of a generalized Heisenberg group [9, § R ξ U = 0 , R ξ J Z ξ = − J Z ξ, R ξ Z = 14 Z, for every U ∈ ( J ξ ) ⊥ and Z ∈ z .Inserting (4), (6) and (7) into (1), we get that ( J ξ ) ⊥ , J ξ and z are invariant subspacesfor the Ricci operator Ric of the hypersurface S . Moreover Ric is given by(8) Ric | ( J ξ ) ⊥ = − m , Ric | J ξ = 1 − m , Ric | z = n −
24 Id . At this point, we will organize the arguments towards the proof of Theorem B intoseveral propositions that will allow us to analyze in which cases S is an algebraic Riccisoliton. Thus, we start with the following result. Proposition 3.1. S is an algebraic Ricci soliton if and only if at least two of the followingthree conditions hold: J ξ is abelian, ( J ξ ) ⊥ is abelian, and [ J ξ, ( J ξ ) ⊥ ] = 0 .Proof. First note that S is an algebraic Ricci soliton if and only if there exists c ∈ R suchthat the endomorphism D := Ric − c Id of s , which in view of (8) is given by(9) D | ( J ξ ) ⊥ = − (cid:16) m c (cid:17) Id , D | J ξ = (cid:18) − m − c (cid:19) Id , D | z = (cid:18) n − − c (cid:19) Id , is a derivation of s , that is, for any U, V ∈ s it satisfies(10) D [ U, V ] = [
DU, V ] + [
U, DV ] . Given any U , V ∈ J ξ , we have D [ U, V ] = (cid:18) n − − c (cid:19) [ U, V ] , [ DU, V ] = [
U, DV ] = (cid:18) − m − c (cid:19) [ U, V ] . Then (10) holds for every
U, V ∈ J ξ if and only if c = (6 − m − n ) / J ξ is abelian.Arguing similarly, we deduce that (10) holds for every U, V ∈ ( J ξ ) ⊥ if and only if c =(2 − m − n ) / J ξ ) ⊥ is abelian; and (10) holds for every U ∈ J ξ and V ∈ ( J ξ ) ⊥ if andonly if c = (4 − m − n ) / J ξ, ( J ξ ) ⊥ ] = 0.Therefore, if S is an algebraic Ricci soliton (i.e. D ∈ Der( s )), then at least two of theconditions in the statement hold (since otherwise we would get two different values for c ).Conversely, since z is the center of n , and D leaves both v ⊖ R ξ and z invariant, it turns outthat, for a fixed c ∈ R , D ∈ Der( s ) if and only if (10) holds for all U , V ∈ J ξ ⊕ ( J ξ ) ⊥ . But,in view of the equivalences in the previous paragraph, if at least two of the conditions in thestatement are satisfied, then there exists c ∈ R satisfying (10) for all U , V ∈ J ξ ⊕ ( J ξ ) ⊥ . (cid:3) Note that, from the definition of J : z → End( v ), we deduce that, given U , V ∈ v , we have[ U, V ] = 0 if and only if h J Z U, V i = 0 for all Z ∈ z . Hence, a subspace w of v is abelian ifand only if Jw ⊥ w . It will be helpful to take this remark into account in what follows. Proposition 3.2. ( J ξ ) ⊥ is abelian if and only if one of the following conditions holds: (i) dim( J ξ ) ⊥ ∈ { , } , (ii) N is isomorphic to N ( m, , where m ∈ { , } , and ξ belongs to one of the two m -dimensional irreducible half-spin submodules of the Spin ( m ) -module v .Moreover, in case (ii) , J ξ is abelian.Proof. Assume that ( J ξ ) ⊥ is abelian. Then, if U ∈ ( J ξ ) ⊥ we have h J Z U, V i = h [ U, V ] , Z i =0 for each V ∈ ( J ξ ) ⊥ and Z ∈ z . Moreover, h J Z U, ξ i = −h J Z ξ, U i = 0. Hence, J U ⊂ J ξ ,for each U ∈ ( J ξ ) ⊥ . In particular, for any fixed non-zero Z ∈ z , we have that J Z maps( J ξ ) ⊥ injectively into J ξ . Therefore, n − m − J ξ ) ⊥ ≤ dim J ξ = m , which impliesthat n ≤ m + 1. But according to the classification of generalized Heisenberg groupsand their dimensions (or, equivalently, according to the classification of Clifford modulesover Cl ( m ), see Table 1), there is only a finite number of groups satisfying such inequality,namely the groups with 1 ≤ m ≤ k = 1. For m ∈ { , , } we have n = m + 1 andhence dim( J ξ ) ⊥ = 0, whereas for m ∈ { , } we have n = m + 2 and then dim( J ξ ) ⊥ = 1.We are left with the cases m ∈ { , , } .For N (5 ,
1) we have n = 8, whence dim( J ξ ) ⊥ = 2. If we restrict the Clifford repre-sentation of Cl (5) on v to a Cl (3)-module, then v turns out to be isomorphic to H ⊕ H .Since the action of Spin (5) ∼ = Sp (2) on v is transitive on the unit sphere of v , we mayassume (by conjugating the Clifford subalgebra Cl (3) by an element of Spin (5) if neces-sary) that ξ is contained in one of these H -factors, and then ( J ξ ) ⊥ is contained in theother H -factor. In other words, v = R ξ ⊕ e J ξ ⊕ R U ⊕ e J U , where we take U ∈ ( J ξ ) ⊥ and e J = span { J Z , J Z , J Z } for some orthonormal subset Z , Z , Z of z . Then there exists J ∈ e J such that ( J ξ ) ⊥ = R U ⊕ R J U , from where we deduce that ( J ξ ) ⊥ is not abelian,which contradicts our initial assumption.In order to finish the proof of the necessity of the proposition we have to analyze the cases N ( m, m ∈ { , } . Note that in both cases n = dim v = 2 m , whence dim( J ξ ) ⊥ = m − N = N ( m, m ∈ { , } . Since ( J ξ ) ⊥ is abelian by assumption, then foreach non-zero J ∈ J we have J ( J ξ ) ⊥ ⊥ ( J ξ ) ⊥ , that is, J ( J ξ ) ⊥ ⊂ R ξ ⊕ J ξ . Here one caneasily see that ξ and J ξ are perpendicular to J ( J ξ ) ⊥ , and hence J ( J ξ ) ⊥ ⊂ J ξ ⊖ R J ξ ; butsince both subspaces have the same dimension m −
1, we have J ( J ξ ) ⊥ = J ξ ⊖ R J ξ . Thenalso J ( J ξ ⊖ R J ξ ) = ( J ξ ) ⊥ , for any non-zero J ∈ J . Take an orthonormal basis { Z , . . . , Z m } of z and put J i = J Z i , i ∈ { , . . . , m } . Then { J i J j ξ : 1 ≤ i < j ≤ m } spans ( J ξ ) ⊥ . Considerthe Spin ( m )-action on v induced by the Cl ( m )-module v . On the one hand, this action isknown to be the sum of the two irreducible half-spin representations, v = ∆ + ⊕ ∆ − (see forexample [23, Proposition I.5.12]). This representation restricts to the unit sphere S m − of v , and the resulting action is of cohomogeneity one (see [3, Table 1] for m = 8, whereasif m = 4 then Spin (4) ∼ = Sp (1) × Sp (1) acts factorwise on v ∼ = H ⊕ H ). In fact, the orbit Spin ( m ) · ξ is singular if and only if ξ is in one of the irreducible components ∆ + and ∆ − . In this case the orbit is isometric to the sphere S m − , to be exact, ( Sp (1) × Sp (1)) / Sp (1) ∼ = S or Spin (8) / Spin (7) ∼ = S . Note that the regular orbits are Sp (1) × Sp (1) ∼ = S × S and Spin (8) / G ∼ = S × S , which are of codimension one in the unit sphere of v . On the otherhand, we can compute the tangent space to the orbit through ξ as T ξ ( Spin ( m ) · ξ ) = spin ( m ) · ξ = span { J i J j ξ : 1 ≤ i < j ≤ m } = ( J ξ ) ⊥ . Hence,
Spin ( m ) · ξ is ( m − ξ ∈ ∆ + or ξ ∈ ∆ − . This completes one of the implications inthe statement.Let us now show the converse. If dim( J ξ ) ⊥ ∈ { , } , then ( J ξ ) ⊥ is trivially abelian.So let us assume that N is isomorphic to N ( m,
1) with m ∈ { , } . The decomposition v = ∆ + ⊕ ∆ − coincides with the ( ± ω = J · · · J m (see [23, Proposition I.5.10]), and then any non-zero J ∈ J interchanges ∆ + and ∆ − (see [23, Proposition I.3.6]). Hence, since v = ∆ + ⊕ ∆ − is an orthogonal directsum, we get that ∆ + and ∆ − are abelian. Now, if ξ ∈ ∆ + then J ξ = ∆ − . This proves thelast claim of the statement, but it also implies that ( J ξ ) ⊥ = ∆ + ⊖ R ξ , and hence ( J ξ ) ⊥ isabelian. We argue analogously if ξ ∈ ∆ − , and this concludes the proof. (cid:3) Remark . Notice that dim( J ξ ) ⊥ = 0 if and only if N is isomorphic to N (1 , N (3 , , N (7 , , J ξ ) ⊥ = 1 corresponds precisely to N (2 ,
1) and N (6 , N is isomorphic to N (2 ,
1) then J ξ is abelian. Indeed, in this case, if { Z , Z } is anorthonormal basis of z , and J i := J Z i , then h [ J i ξ, J i +1 ξ ] , Z i i = h J i ξ, J i +1 ξ i = 0, for each i ∈ { , } . However, if N is isomorphic to N (6 ,
1) then J ξ is not abelian, since dim v = 8,dim J ξ = 6, and hence J ( J ξ ) and J ξ must intersect non-trivially, for any non-zero J ∈ J .Moreover, in this case, [ J ξ, ( J ξ ) ⊥ ] = 0, since dim J ( J ξ ) ⊥ = 6 = dim J ξ and therefore J ( J ξ ) ⊥ and J ξ have non-trivial intersection. Proposition 3.4. If [ J ξ, ( J ξ ) ⊥ ] = 0 and m ≥ , then J ξ is not abelian.Proof. Let V ∈ ( J ξ ) ⊥ , W ∈ J ξ and Z ∈ z . Then, by assumption, 0 = h [ W, V ] , Z i = h J Z W, V i , which implies that J ( J ξ ) ⊂ R ξ ⊕ J ξ . Fix any non-zero J ∈ J . Hence, bydimension reasons, we have J ( J ξ ) = ( J ξ ⊖ R J ξ ) ⊕ R ξ . Therefore, J ξ ⊕ R ξ is a J -complexsubspace of v , where we endow v with the complex structure J . Hence, if m ≥ J ξ admitsa J -complex subspace of the form R W ⊕ R J W , with W = 0, and then [ W, J W ] = 0 by (2),which concludes the proof. (cid:3) At this point, we can finish the proof of the main result of this section.
Proof of Theorem B.
First observe that, by [21], a simply connected nilpotent Lie groupwith a left-invariant metric is an algebraic Ricci soliton if and only if it is a Ricci soliton.Since S is simply connected and nilpotent (as a codimension one subgroup of N ), in orderto prove the theorem we just have to analyze the algebraic Ricci soliton condition.By Proposition 3.1, the four items in the statement of Theorem B give rise to algebraicRicci solitons. Indeed, in case (i) J ξ is one-dimensional and, hence, abelian, and also[ J ξ, ( J ξ ) ⊥ ] = 0 since J ( J ξ ) = R ξ is orthogonal to ( J ξ ) ⊥ . In case (ii), ( J ξ ) ⊥ is one-dimensional and J ξ is abelian as well (see Remark 3.3). In case (iii) we have ( J ξ ) ⊥ = 0 (again as noticed in Remark 3.3), and then [ J ξ, ( J ξ ) ⊥ ] = 0. Finally, in case (iv), both J ξ and ( J ξ ) ⊥ are abelian by Proposition 3.2.In order to prove the converse, assume that S is an algebraic Ricci soliton, and let usapply again Proposition 3.1. We distinguish two main cases depending on whether ( J ξ ) ⊥ is abelian or not.Assume first that ( J ξ ) ⊥ is abelian. Then, by Proposition 3.2, either dim( J ξ ) ⊥ ∈ { , } or N corresponds to item (iv) in the statement of Theorem B. If ( J ξ ) ⊥ = 0 (and hencealso [ J ξ, ( J ξ ) ⊥ ] = 0), then, as noted in Remark 3.3, N is isomorphic to N (1 , N (3 , , N (7 , , J ξ ) ⊥ = 1, then N is isomorphic to N (2 ,
1) or N (6 , J ξ is abelian (and this leads to item (ii) in the statement), whereas in the second case J ξ is not abelian and [ J ξ, ( J ξ ) ⊥ ] = 0.Finally, assume that ( J ξ ) ⊥ is not abelian. Therefore, by Proposition 3.1 we must havethat J ξ is abelian and [ J ξ, ( J ξ ) ⊥ ] = 0. Then, Proposition 3.4 implies that N is isomorphicto N (1 , k ), which leads to item (i) in the statement of Theorem B. (cid:3) Solvsoliton hypersurfaces of Damek-Ricci spaces
This section is devoted to derive the classification of codimension one Ricci soliton sub-groups of Damek-Ricci spaces. In § § Damek-Ricci spaces.
Let a be a one-dimensional real vector space, B ∈ a a non-zero vector, and n a generalizedHeisenberg algebra. We will use the notations and facts introduced in § n = v ⊕ z . We define the vectorspace direct sum a ⊕ n , which we endow with the Lie algebra structure determined by theLie algebra structure of the generalized Heisenberg algebra n and the relations[ B, U ] = 12 U, [ B, Z ] = Z, for every U ∈ v , Z ∈ z . This converts a ⊕ n into a solvable Lie algebra. We equip a ⊕ n with the positive definiteinner product h· , ·i that extends the inner product of n , makes B into a unit vector and a ⊕ n into an orthogonal direct sum. The simply connected solvable Lie group AN with Liealgebra a ⊕ n , endowed with the left-invariant Riemannian metric determined by the innerproduct h· , ·i on a ⊕ n , is called a Damek-Ricci space . Analogously as in § AN ( m, k ) or AN ( m, k + , k − ) to refer to a Damek-Ricci space whose underlyinggeneralized Heisenberg group with Lie algebra n is N ( m, k ) or N ( m, k + , k − ), respectively.Damek-Ricci spaces constitute an important family of Einstein solvmanifolds. This fam-ily contains the rank one symmetric spaces of non-compact type and non-constant curva-ture, that is, the hyperbolic spaces over the complex numbers, the quaternions and the octo-nions. Indeed, these are the only symmetric Damek-Ricci spaces: AN (1 , k ), k ≥
1, which ishomothetic to a complex hyperbolic space C H k +1 , AN (3 , k, ∼ = AN (3 , , k ), k ≥
1, whichis homothetic to a quaternionic hyperbolic space H H k +1 , and AN (7 , , ∼ = AN (7 , , which is homothetic to the Cayley hyperbolic plane O H . The real hyperbolic spaces R H k +1 , k ≥
1, could be considered as members of this family if one had allowed z = 0 (in this case n = v would be abelian). However, in this section we assume that dim z >
0, as our problemin real hyperbolic spaces is straightforward (see the proof of Theorem A at the end of § AN is given by the formula(11) ∇ sB + V + Y (cid:0) rB + U + X (cid:1) = (cid:18) h U, V i + h X, Y i (cid:19) B − J X V − J Y U − rV −
12 [
U, V ] − rY, for any s, r ∈ R , U, V ∈ v and X, Y ∈ z . Using this, one can derive formulas for thecurvature tensor and, thus, one can deduce that the Ricci operator is given by(12) Ric AN = − (cid:16) m + n (cid:17) Id , where we recall that m = dim z and n = dim v . For more information on Damek-Riccispaces, we refer to [9].4.2. Proof of the classification result.
Let S be a connected Lie subgroup of codimension one of a Damek-Ricci space AN . Let s = ( a ⊕ n ) ⊖ R ξ be the corresponding Lie subalgebra of a ⊕ n , where ξ = aB + U + Z is aunit vector, with a ∈ R , U ∈ v and Z ∈ z . Then J Z U , | Z | U − | U | Z ∈ s ∩ n and, by (5)(with W = U ) and [ v , v ] = z , we deduce that Z = 0. Therefore ξ = aB + U, with a ∈ R , U ∈ v , a + | U | = 1 . Note that, in this case, we have the orthogonal decompositions s = R ( | U | B − aU ) ⊕ ( v ⊖ R U ) ⊕ z = R ( | U | B − aU ) ⊕ J U ⊕ ( J U ) ⊥ ⊕ z , where, as in § J = { J Z : Z ∈ z } , J U = { J Z U : Z ∈ z } and ( J U ) ⊥ = v ⊖ ( R U ⊕ J U ).Let us calculate the shape operator of S by making use of (11): S ξ ( | U | B − aU ) = −∇ | U | B − aU ( aB + U ) = a | U | B − aU ) , S ξ V = −∇ V ( aB + U ) = a V, for any V ∈ ( J U ) ⊥ , S ξ J Z U = −∇ J Z U ( aB + U ) = 12 ( aJ Z U + | U | Z ) , for any Z ∈ z , J Z U ∈ J U, (13) S ξ Z = −∇ Z ( aB + U ) = 12 J Z U + aZ, for any Z ∈ z . In particular(14) tr S ξ = a a n − m −
1) + a m + am = a (cid:18) m + n (cid:19) . In order to calculate the normal Jacobi operator R ξ = R AN ( · , ξ ) ξ of S , we can combinethe definition of the curvature tensor of the ambient space, R AN ( X, Y ) = [ ∇ X , ∇ Y ] −∇ [ X,Y ] , with the formula (11) for the Levi-Civita connection, or alternatively use the formula forthe Jacobi operator of a Damek-Ricci space [9, § R ξ ( | U | B − aU ) = −
14 ( | U | B − aU ) ,R ξ V = − V, for any V ∈ ( J U ) ⊥ ,R ξ J Z U = − (cid:0) | U | + 1 (cid:1) J Z U − a | U | Z, for any Z ∈ z , J Z U ∈ J U,R ξ Z = − aJ Z U + (cid:18) | U | − (cid:19) Z, for any Z ∈ z . Therefore, by inserting (12), (13), (14) and (15) into (1), recalling that a + | U | = 1, andsetting for convenience(16) e m := − m − n e n := m + n , we obtain that the Ricci operator of the hypersurface S is given by:Ric( V ) = 14 (cid:0) | U | + 4 e m + 2 a e n (cid:1) V, Ric( J Z U ) = 14 (cid:0) | U | + 4 e m + 2 a e n (cid:1) J Z U + e n a | U | Z, Ric( Z ) = e n aJ Z U + ( e m + a e n ) Z, (17)for any V ∈ R ( | U | B − aU ) ⊕ ( J U ) ⊥ and Z ∈ z .In what follows we will study when S is an algebraic Ricci soliton, which amounts todetermining under which circumstances and for which values of c ∈ R the endomorphism D := Ric − c Id of s is a derivation of s . Proposition 4.1. If S is an algebraic Ricci soliton, then ξ ∈ a or ξ ∈ v . Moreover, if ξ ∈ v then c = (1 + 4 e m ) / .Proof. If S is an algebraic Ricci soliton, then D must, in particular, satisfy D [ | U | B − aU, Z ] = [ D ( | U | B − aU ) , Z ] + [ | U | B − aU, DZ ] , for any Z ∈ z . By the definition of D and (17), the term on the left-hand side turns outto be D [ | U | B − aU, Z ] = | U | DZ = | U | e n aJ Z U + | U | (cid:0) e m + a e n − c (cid:1) Z. Similarly, for the addends of the right-hand side we have: (cid:2) D ( | U | B − aU ) , Z (cid:3) = 14 (cid:0) | U | + 4 e m + 2 a e n − c (cid:1) | U | Z, (cid:2) | U | B − aU, DZ (cid:3) = e n a | U | J Z U + | U | (cid:18) e m − c + e n a (cid:19) Z. Altogether, considering the component in the direction of J Z U , we get from the firstequation of the proof that either U = 0 or a = 0, which proves the first claim in thestatement. Finally, if ξ ∈ v , then a = 0 and | U | = 1, and taking the Z -component in theequation considered above we get the expression for c in the statement. (cid:3) Note that, if ξ = U ∈ v , then (17) translates into(18) Ric | a ⊕ ( J ξ ) ⊥ = (cid:18) e m + 14 (cid:19) Id , Ric | J ξ = (cid:18) e m + 34 (cid:19) Id , Ric | z = e m Id . We can now conclude the proof of the main result of this section.
Proof of Theorem C.
First note that, since a ⊕ n is completely solvable (i.e. ad( X ) has onlyreal eigenvalues, for each X ∈ a ⊕ n ), the same is true for the Lie subalgebra s = ( a ⊕ n ) ⊖ R ξ .Hence, by [22], the connected, simply connected Lie subgroup S of a Damek-Ricci space AN with Lie algebra s is a Ricci soliton if and only if it is an algebraic Ricci soliton.Now, observe that the examples mentioned in the statement are indeed algebraic Riccisolitons. On the one hand, any generalized Heisenberg group N is an algebraic Riccisoliton, as remarked in § S is a Lohnherr hypersurface in C H ,then it follows from (17) that D = Ric − c Id, with c = (1 + 4 e m ) /
4, has eigenvalues 0, 1 / − / a , J ξ and z . One can easily check that D is a derivation of s = a ⊕ ( v ⊖ R ξ ), which shows that S is an algebraic Ricci soliton.Finally, we prove the converse implication. Thus, assume that S is an algebraic Riccisoliton. By Proposition 4.1 we have that ξ ∈ a or ξ ∈ v .If ξ ∈ a , then s = n is a generalized Heisenberg algebra. Hence, the correspondingconnected Lie group S coincides with the generalized Heisenberg group N .Let us assume that ξ ∈ v . By Proposition 4.1 we have c = (1 + 4 e m ) /
4. Then by (18)we have that D = Ric − c Id satisfies D | a ⊕ ( J ξ ) ⊥ = 0 , D | J ξ = 12 Id , D | z = −
14 Id . Since D is a derivation, one can see that ( J ξ ) ⊥ and J ξ are abelian subspaces of v , and[( J ξ ) ⊥ , J ξ ] = 0. But then, by Proposition 3.4 we must have m = dim z = 1, which impliesthat AN is isometric to a complex hyperbolic space. Moreover, by Proposition 3.2 we alsohave dim( J ξ ) ⊥ ∈ { , } , which forces AN to be in fact isometric to a complex hyperbolicplane C H . Since ξ ∈ v , S is a Lohnherr hypersurface in C H . This concludes the proof. (cid:3) Solvsoliton hypersurfaces of symmetric spaces of non-compact type
This section is devoted to studying which codimension one subgroups of the solvable partof the Iwasawa decomposition of the isometry group of a symmetric space of non-compacttype are Ricci solitons. In § § Symmetric spaces of non-compact type.
Let M be a symmetric space of non-compact type. Then M can be identified with aquotient space G/K , where G is the connected component of the identity of the isometrygroup of M (up to a finite quotient), and K is the isotropy subgroup of G correspondingto some point o ∈ M . Let θ be the corresponding Cartan involution of the Lie algebra g of G , and g = k ⊕ p the associated Cartan decomposition, where k and p are the (+1) and( − θ , respectively. The Lie algebra g is real semisimple, which impliesthat its Killing form B is non-degenerate. Indeed, the Cartan decomposition g = k ⊕ p is B -orthogonal, B | k × k is negative definite, and B | p × p is positive definite (since M is ofnon-compact type). Hence, defining h X, Y i B θ := − B ( θX, Y ) , for all X, Y ∈ g , we have that h· , ·i B θ is a positive definite inner product on g . It is easy to check that thisinner product satisfies(19) h ad( X ) Y, Z i B θ = −h Y, ad( θX ) Z i B θ , for all X, Y, Z ∈ g . Let a be a maximal abelian subspace of p . The rank of M is precisely the dimension of a .Then { ad( H ) : H ∈ a } constitutes a commuting family of self-adjoint endomorphisms of g , and hence they diagonalize simultaneously. Their common eigenspaces are called therestricted root spaces of g , whereas their non-zero eigenvalues (which depend linearly on H ∈ a ) are the restricted roots of g . In other words, if for each covector λ ∈ a ∗ we define g λ := { X ∈ g : [ H, X ] = λ ( H ) X for all H ∈ a } , then any g λ = 0 is a restricted root space, and any λ = 0 such that g λ = 0 is a restrictedroot. Notice that g is always non-zero, since a ⊂ g . If Σ = { λ ∈ a ∗ : λ = 0 , g λ = 0 } denotes the set of restricted roots, then the h· , ·i B θ -orthogonal decomposition g = g ⊕ (cid:18)M λ ∈ Σ g λ (cid:19) is called the restricted root space decomposition of g . Moreover, we have the bracketrelations [ g λ , g µ ] ⊂ g λ + µ and θ g λ = g − λ , for any λ , µ ∈ a ∗ . We also have the h· , ·i B θ -orthogonal decomposition g = k ⊕ a , where k = g ∩ k is the normalizer of a in k .For each λ ∈ Σ, we define H λ ∈ a by the relation B ( H λ , H ) = λ ( H ), for all H ∈ a .We can introduce an inner product on a ∗ given by h λ, µ i := B ( H λ , H µ ). We will write | λ | = h λ, λ i / for the induced norm on a ∗ . Then, we have that Σ is an abstract rootsystem in a ∗ . Thus, we can define a positivity criterion on a ∗ , which allows us to decomposeΣ = Σ + ∪ ( − Σ + ), where Σ + is the set of positive roots. We will denote by Π ⊂ Σ + thecorresponding set of simple roots, that is, those positive roots λ ∈ Σ + which are not sumof two positive roots. Then, any root λ ∈ Σ is a linear combination of elements of Π withinteger coefficients, where all of them are either non-negative (if λ ∈ Σ + ) or non-positive(if λ ∈ − Σ + ). We will also make use of the so-called Cartan integers, that is, the integers A α,λ := 2 h α, λ i| α | , for each α , λ ∈ Σ. We refer to [19] for more details on root systems.A fundamental result for our work is the Iwasawa decomposition theorem. At the Liealgebra level, it states that g = k ⊕ a ⊕ n is a vector space direct sum, where n = L λ ∈ Σ + g λ .Note that n and a ⊕ n are nilpotent and solvable Lie subalgebras of g , respectively; indeed[ a ⊕ n , a ⊕ n ] ⊂ n . Let A , N and AN denote the (closed) connected Lie subgroups of G with Lie algebras a , n and a ⊕ n , respectively. The Iwasawa decomposition at the Liegroup level states that G is diffeomorphic to the Cartesian product K × A × N . It followsthat AN acts simply transitively on M ∼ = G/K , and hence M is diffeomorphic to thesolvable Lie group AN . If we pull back the symmetric Riemannian metric on M to AN via this diffeomorphism, we obtain a left-invariant metric on AN . Thus, any symmetricspace of non-compact type M ∼ = G/K is isometric to the solvable part AN of an Iwasawadecomposition for G , equipped with a certain left-invariant metric. We will denote by h· , ·i this left-invariant metric on AN , and also the corresponding inner product on a ⊕ n .In what follows, the symmetric space M will be assumed to be irreducible. If X, Y ∈ a ⊕ n ,and denoting orthogonal projections (with respect to h· , ·i B θ ) with subscripts, the followingrelation between h· , ·i and h· , ·i B θ holds (up to homothety of the metric on M ):(20) h X, Y i = h X a , Y a i B θ + 12 h X n , Y n i B θ . Using Koszul’s formula and relations (20)-(19), one can obtain a formula for the Levi-Civitaconnection ∇ of the Lie group AN with its left-invariant metric. Indeed, if X, Y, Z ∈ a ⊕ n :(21) h∇ X Y, Z i = 14 h [ X, Y ] + [ θX, Y ] − [ X, θY ] , Z i B θ . We warn of the use of two different inner products on the previous formula.We end this subsection with two lemmas that will be useful in what follows. We alsonote that, from now on, whenever we consider a unit vector X ∈ a ⊕ n we will understandthat it has length one with respect to the inner product h· , ·i defined on a ⊕ n . Lemma 5.1. [6, Lemma 2.3]
Let λ ∈ Σ + and X, Y ∈ g λ be orthogonal. Then: (i) [ θX, X ] = 2 h X, X i H λ = h X, X i B θ H λ . (ii) [ θX, Y ] ∈ k = g ⊖ a . Lemma 5.2.
Let ξ = aH α + bX α be a unit vector, where α ∈ Π is a simple root, X α ∈ g α is a unit vector, and a , b ∈ R . Then: (i) [ θξ, ξ ] = − ab | α | X α + ab | α | θX α + 2 b H α . (ii) ∇ ξ ξ = b H α − ab | α | X α . (iii) ∇ H X = 0 for all H ∈ a and X ∈ a ⊕ n .Proof. First, using the facts that a is abelian, θ | a = − Id, θ g λ = g − λ for all λ ∈ Σ, thedefinition of restricted root space, and Lemma 5.1 (i), we deduce[ θξ, ξ ] = [ θ ( aH α + bX α ) , aH α + bX α ] = − ab [ H α , X α ] + ab [ θX α , H α ] + b [ θX α , X α ]= − ab | α | X α + ab | α | θX α + 2 b H α , which proves assertion (i). Now, if Z ∈ a ⊕ n , using (21), assertion (i), and (20), we get h∇ ξ ξ, Z i = 14 h [ ξ, ξ ] + [ θξ, ξ ] − [ ξ, θξ ] , Z i B θ = 12 h [ θξ, ξ ] , Z i B θ = 12 h− ab | α | X α + ab | α | θX α + 2 b H α , Z i B θ = − ab | α | h X α , Z i B θ + b h H α , Z i B θ = − ab | α | h X α , Z i + b h H α , Z i . This proves assertion (ii). Finally, using again (21), we have h∇ H X, Z i = 14 h [ H, X ] + [ θH, X ] − [ H, θX ] , Z i B θ = 0 , for all X , Z ∈ a ⊕ n and all H ∈ a . This finishes the proof. (cid:3) Proof of the classification result.
Let M ∼ = G/K be an irreducible symmetric space of non-compact type, and let AN be the solvable part of an Iwasawa decomposition of G . Let S be a codimension onesubgroup of AN with Lie algebra s . Then there exists a unit vector ξ ∈ a ⊕ n such that s = ( a ⊕ n ) ⊖ R ξ . We have (see [7, Lemma 5.3]): Lemma 5.3. If s = ( a ⊕ n ) ⊖ R ξ is a Lie subalgebra of a ⊕ n , then ξ ∈ a or ξ ∈ R H α ⊕ g α for some simple root α ∈ Π . The subgroup S of AN , with the induced metric, is by definition an algebraic Riccisoliton if the Ricci tensor Ric of S can be written as Ric = c Id + D , where c ∈ R and D is a derivation of s . Since an irreducible symmetric space of non-compact type is Einsteinwith negative scalar curvature, the Ricci operator of M satisfies Ric M = k Id for some k <
0. Therefore, by (1), studying when S is an algebraic Ricci soliton is equivalent todetermining when the endomorphism(22) D := tr( S ξ ) S ξ − ( R ξ + S ξ ) + c Idof s is a derivation, for some c ∈ R .Lemma 5.3 indicates that we have to consider two cases. The first one (correspondingto ξ ∈ a ) has been addressed in [11, Proposition 5.2], but we include a direct proof herefor the sake of completeness. Remark . The orbits of the connected Lie subgroup S of AN with Lie algebra s = ( a ⊖ R H ) ⊕ n , for some unit vector H in the Weyl chamber { H ∈ a : λ ( H ) ≥ λ ∈ Σ + } ,are horospheres of the symmetric space M , meaning that they are level sets of a Busemannfunction [13, § H is a geodesic vector field.Let γ be one of the geodesic integral curves of H . As H is left-invariant, any other integralcurve of H is of the form g ◦ γ , for some g ∈ AN . The group AN is contained in the parabolicsubgroup of G given by the stabilizer of the point at infinity lim t →∞ γ ( t ) in the ideal bound-ary of M (see [13, § H are geodesics asymptotic to suchcommon point at infinity. Consider the Busemann function f H ( p ) = lim t → + ∞ ( d ( p, γ ( t )) − t )on M associated with γ . By [13, § f H = − H everywhere. Thus, the level sets of f H are everywhere orthogonal to H , which implies that they are preciselythe S -orbits. Finally, note that the S -orbits are mutually congruent (cf. [7]). Proposition 5.5.
Let M be an irreducible symmetric space of non-compact type withsolvable Iwasawa group AN . Let S be the connected Lie subgroup of AN with Lie algebra s = ( a ⊖ R H ) ⊕ n , for some unit vector H ∈ a . Then S is a solvsoliton.Proof. Let X ∈ n and Y ∈ a ⊕ n . Then, using (21) and (20) we have h∇ X H, Y i = 14 h [ X, H ] + [ θX, H ] − [ X, θH ] , Y i B θ = − h [ H, X ] , Y i B θ = −h [ H, X ] , Y i . This means that S H X = ad( H ) X for all X ∈ n . Now, using this and Lemma 5.2 (iii)twice, we have R H ( X ) = ∇ X ∇ H H − ∇ H ∇ X H − ∇ [ X,H ] H = −S H [ H, X ] = − ad( H ) X, for all X ∈ n . Moreover, using Lemma 5.2 (iii) we easily get S H | a ⊖ R H = R H | a ⊖ R H = 0. Allin all, we deduce S H = ad( H ) and R H = − ad( H ) . Therefore, we getRic = Ric M + tr( S H ) S H − R H − S H = k Id + tr(ad( H )) ad( H ) , which is precisely the sum of a derivation of s and a multiple of the identity. (cid:3) From now one we will focus on dealing with the second case in Lemma 5.3. Moreprecisely, the rest of this section will be devoted to prove the following:
Theorem 5.6.
Let ξ = aH α + bX α be a unit vector, where X α ∈ g α is a unit vector, α ∈ Π a simple root, and a , b ∈ R , b = 0 . Let S be the connected Lie subgroup of AN with Liealgebra s = ( a ⊕ n ) ⊖ R ξ . If the hypersurface S , with the induced metric, is an algebraicRicci soliton, then dim a = 1 , that is, M is a rank one symmetric space. Thus, we will assume from now on that M is an irreducible symmetric space of non-compact type and rank greater than one, s = ( a ⊕ n ) ⊖ R ξ is a Lie subalgebra of a ⊕ n with ξ = aH α + bX α for some simple root α ∈ Π, some unit vector X α ∈ g α , and a , b ∈ R .Note that, since ξ and X α are unit vectors, we also have that a | α | + b = 1. Our goal willbe to prove that, if S is an algebraic Ricci soliton (that is, if the endomorphism D givenin (22) is a derivation), then b = 0.Let us consider the vector U := b | α | − H α − a | α | X α . Note that h U, U i = 1 and h U, ξ i = 0. Thus, we can consider the h· , ·i -orthogonal decompo-sition s = ( a ⊖ R H α ) ⊕ ( n ⊖ R X α ) ⊕ R U of the tangent space to S at the identity.Our efforts will be focused on calculating the endomorphism D of s . Although onecould determine it completely, it will be enough for our purposes to just calculate D whenrestricted to a particular subspace. Actually, we will select a root β ∈ Σ + \{ α } such that β + α ∈ Σ and β − α / ∈ Σ. Such a root exists since M is irreducible and has rank greaterthan one by assumption: it suffices to choose a simple root β connected to α in the Dynkin diagram of M . Then, the main part of what follows will be devoted to determining therestriction of the endomorphism D to the subspace(23) ( a ⊖ R H α ) ⊕ ( g β ⊕ g β + α ) ⊕ R U ⊂ s . We start by analyzing the restriction of D to a ⊖ R H α . Proposition 5.7.
Let H ∈ a ⊖ R H α . Then DH = cH .Proof. Let H ∈ a ⊖ R H α be arbitrary. From Lemma 5.2 (iii) we obtain S ξ H = −∇ H ξ = 0,and hence S ξ | a ⊖ R H α = S ξ | a ⊖ R H α = 0. Note now that [ H, ξ ] = [
H, aH α + bX α ] = 0, since H ∈ a ⊖ R H α . Using this and Lemma 5.2 (iii) twice, we get R ξ ( H ) = R AN ( H, ξ ) ξ = ∇ H ∇ ξ ξ − ∇ ξ ∇ H ξ − ∇ [ H,ξ ] ξ = 0for the Jacobi operator. All in all, from (22) we conclude DH = (tr( S ξ ) S ξ − R ξ − S ξ + c Id) H = cH. (cid:3) We have just determined D | a ⊖ R H α . Before continuing with the calculation of D , we willintroduce two results that will allow us to use the Levi-Civita connection more efficiently. Lemma 5.8.
Let X = X a + P γ ∈ Σ + X γ and Y = Y a + P γ ∈ Σ + Y γ be vectors in a ⊕ n , with X a , Y a ∈ a and X γ , Y γ ∈ g γ for all γ ∈ Σ + . If h X γ , Y γ i = 0 for all γ ∈ Σ + , then ∇ X Y = 12 ([ X, Y ] + [ θX, Y ] − [ X, θY ]) n , where ( · ) n denotes the orthogonal projection onto the Lie subalgebra n .Proof. First of all, using (19) and the assumption that h X γ , Y γ i = 0 for all γ ∈ Σ + , we have h [ X, θY ] , H i B θ = h X, [ Y, H ] i B θ = −h X, [ H, Y a + X γ ∈ Σ + Y γ ] i B θ = − X γ ∈ Σ + γ ( H ) h X, Y γ i B θ = − X γ ∈ Σ + γ ( H ) h X a , Y γ i B θ − X γ ∈ Σ + X λ ∈ Σ + γ ( H ) h X λ , Y γ i B θ = 0 , for all H ∈ a . This means that [ X, θY ] ∈ g is h· , ·i B θ -orthogonal to a . Analogously,[ θX, Y ] ∈ g is also h· , ·i B θ -orthogonal to a . Since [ X, Y ] ∈ [ a ⊕ n , a ⊕ n ] = n , then[ X, Y ] + [ θX, Y ] − [ X, θY ] is h· , ·i B θ -orthogonal to a . Now, using this, together with (21)and (20), we deduce h∇ X Y, Z i = 14 h [ X, Y ] + [ θX, Y ] − [ X, θY ] , Z i B θ = 14 h ([ X, Y ] + [ θX, Y ] − [ X, θY ]) n , Z n i B θ = 12 h ([ X, Y ] + [ θX, Y ] − [ X, θY ]) n , Z i for all Z ∈ a ⊕ n . This concludes the proof. (cid:3) Lemma 5.9.
Let ξ = aH α + bX α , where α ∈ Π , X α ∈ g α , and a , b ∈ R . Let Y λ ∈ g λ be avector orthogonal to ξ , for some λ ∈ Σ + . Then: (i) h [ θY λ , ξ ] , Z i B θ = h [ θY λ , X α ] , Z i B θ = 0 for all Z ∈ a ⊕ n . (ii) h [ θ [ Y λ , ξ ] , ξ ] , Z i B θ = 0 for all Z ∈ a ⊕ n . (iii) If λ = α , then [ Y λ , θξ ] , [[ Y λ , ξ ] , θξ ] , [ Y λ , θX α ] ∈ n .Proof. First, using the properties of the root space decomposition, we have [ θY λ , ξ ] = a [ θY λ , H α ] + b [ θY λ , X α ] ∈ g − λ ⊕ g α − λ . Since λ is a positive root by assumption, then − λ isnegative, and hence h g − λ , a ⊕ n i B θ = 0. Moreover, since α ∈ Π is a simple root and λ ∈ Σ + ,then α − λ / ∈ Σ + , and hence either λ = α or g α − λ = 0. If λ = α , then [ θY λ , X α ] ∈ k byLemma 5.1 (ii). In any case, [ θY λ , X α ] is h· , ·i B θ -orthogonal to a ⊕ n . This proves claim (i).Now, we have[ θ [ Y λ , ξ ] , ξ ] = [ θ [ Y λ , aH α + bX α ] , ξ ] = − a h α, λ i [ θY λ , ξ ] + b [ θ [ Y λ , X α ] , ξ ] . Note first that [ θY λ , ξ ] is h· , ·i B θ -orthogonal to a ⊕ n by virtue of assertion (i). Moreover,by the properties of the root spaces, we have [ θ [ Y λ , X α ] , ξ ] ∈ g − λ − α ⊕ g − λ , where − λ − α , − λ / ∈ Σ + ∪ { } . This implies assertion (ii).Finally, we prove claim (iii). Again by the properties of the root space decomposition,we have [ Y λ , θξ ] ∈ g λ ⊕ g λ − α and [[ Y λ , ξ ] , θξ ] ∈ g λ ⊕ g λ − α ⊕ g λ + α . Since by assumption λ isa positive root different from α , we have λ + α ∈ Σ + or λ + α / ∈ Σ ∪ { } , and λ − α ∈ Σ + or λ − α / ∈ Σ ∪ { } . In both cases, we get [ Y λ , θξ ], [[ Y λ , ξ ] , θξ ] ∈ n , and taking a = 0 and b = 1 in the first bracket, we also have [ Y λ , θX α ] ∈ n , from where the result follows. (cid:3) Our purpose hereafter will be to calculate the restriction of the endomorphism D definedin (22) to the subspace g β ⊕ g α + β of s (see (23)). This will be done in several steps. Inthe following proposition we restrict our attention to the operator R ξ + S ξ . Note that,since β ∈ Σ + \ { α } and α + β ∈ Σ + by definition of β , for both λ = β and λ = α + β ,we have that λ = α , and hence g λ ⊂ s . Therefore, the following proposition implies that R ξ + S ξ | g β ⊕ g α + β = 0. Proposition 5.10.
Let ξ = aH α + bX α be a unit vector, where α ∈ Π , X α ∈ g α is a unitvector, and a , b ∈ R . Let Y λ ∈ g λ ⊂ s , where λ ∈ Σ + \ { α } . Then ( R ξ + S ξ ) Y λ = 0 .Proof. We will use several times that α − λ / ∈ Σ + , as follows from α ∈ Π and λ ∈ Σ + .First, using Lemma 5.2 (i)-(ii), one has ∇ ξ ξ ∈ a ⊕ g α , ∇ ξ ξ − θ ∇ ξ ξ = [ θξ, ξ ] . Hence, by Lemma 5.8 (taking into account that h Y λ , g α i = 0 since λ = α ), we have ∇ Y λ ∇ ξ ξ = 12 ([ Y λ , ∇ ξ ξ ] + [ θY λ , ∇ ξ ξ ] − [ Y λ , θ ∇ ξ ξ ]) n = 12 [ Y λ , [ θξ, ξ ]] . (24)Now, using again Lemma 5.8, Lemma 5.9 (i)-(iii), and the symmetry of the Levi-Civitaconnection, we get(25) ∇ ξ ∇ Y λ ξ = 12 ∇ ξ ([ Y λ , ξ ] + [ θY λ , ξ ] − [ Y λ , θξ ]) n = 12 ∇ ξ [ Y λ , ξ ] − ∇ ξ [ Y λ , θξ ]= 12 [ ξ, [ Y λ , ξ ]] + 12 ∇ [ Y λ ,ξ ] ξ − ∇ ξ [ Y λ , θξ ] . As λ ∈ Σ + \ { α } , we have that [ Y λ , ξ ] ∈ g λ ⊕ g α + λ and ξ ∈ a ⊕ g α are under the hypothesesof Lemma 5.8. Hence, using Lemma 5.8 together with Lemma 5.9 (ii)-(iii), we have ∇ [ Y λ ,ξ ] ξ = 12 ([[ Y λ , ξ ] , ξ ] + [ θ [ Y λ , ξ ] , ξ ] − [[ Y λ , ξ ] , θξ ]) n = 12 [[ Y λ , ξ ] , ξ ] −
12 [[ Y λ , ξ ] , θξ ] . (26)Again, using Lemma 5.8, Lemma 5.9 (i)-(iii), the symmetry of the Levi-Civita connection,and the Jacobi identity, we obtain(27) S ξ Y λ = ∇ ∇ Yλ ξ ξ = 12 ∇ ([ Y λ ,ξ ]+[ θY λ ,ξ ] − [ Y λ ,θξ ]) n ξ = 12 ∇ [ Y λ ,ξ ] ξ − ∇ [ Y λ ,θξ ] ξ = 12 ∇ [ Y λ ,ξ ] ξ + 12 [ ξ, [ Y λ , θξ ]] − ∇ ξ [ Y λ , θξ ]= 12 ∇ [ Y λ ,ξ ] ξ −
12 [ Y λ , [ θξ, ξ ]] −
12 [ θξ, [ ξ, Y λ ]] − ∇ ξ [ Y λ , θξ ] . Now, using (24), (25), (26) and (27) we get( R ξ + S ξ ) Y λ = ∇ Y λ ∇ ξ ξ − ∇ ξ ∇ Y λ ξ − ∇ [ Y λ ,ξ ] ξ + S ξ Y λ = 12 [ Y λ , [ θξ, ξ ]] −
12 [ ξ, [ Y λ , ξ ]] − ∇ [ Y λ ,ξ ] ξ + 12 ∇ ξ [ Y λ , θξ ] −
12 [[ Y λ , ξ ] , ξ ]+ 12 [[ Y λ , ξ ] , θξ ] + 12 ∇ [ Y λ ,ξ ] ξ −
12 [ Y λ , [ θξ, ξ ]] −
12 [ θξ, [ ξ, Y λ ]] − ∇ ξ [ Y λ , θξ ]= 0 . (cid:3) In view of Proposition 5.10, in order to determine the restriction D | g β ⊕ g α + β of the en-domorphism D defined in (22) to the subspace g β ⊕ g α + β , we just have to calculate therestriction of the shape operator S ξ to g β ⊕ g α + β . Before doing that, we introduce a resultwhich will make easier the calculations concerning the shape operator. Lemma 5.11.
Let Y λ ∈ g λ and X α ∈ g α be unit vectors, where λ ∈ Σ + and α ∈ Π .Assume λ − α / ∈ Σ and λ + α ∈ Σ . Then [ Y λ + α , θX α ] = −| α | p − A α,λ Y λ , where Y λ + α := [ Y λ , X α ] / ( | α | p − A α,λ ) is a unit vector in g λ + α .Proof. First, using (20) taking into account [ Y λ , X α ], Y λ ∈ n , (19), the Jacobi identity, theassumption λ − α / ∈ Σ, Lemma 5.1 (i), and the definition of the Cartan integers A α,λ , weobtain h [ Y λ , X α ] , [ Y λ , X α ] i = 12 h [ Y λ , X α ] , [ Y λ , X α ] i B θ = 12 h Y λ , [ θX α , [ Y λ , X α ]] i B θ = − h Y λ , [ Y λ , [ X α , θX α ]] + [ X α , [ θX α , Y λ ]] i B θ = −h α, λ ih Y λ , Y λ i B θ = −| α | A α,λ h Y λ , Y λ i = −| α | A α,λ . By the standard theory of abstract root systems (see for example [19, Proposition 2.48 (g)])we have A α,λ <
0, as λ − α / ∈ Σ and λ + α ∈ Σ. Hence Y λ + α = [ Y λ , X α ] / ( | α | p − A α,λ ) ∈ g λ + α is a unit vector. Now, using the Jacobi identity, the fact that λ − α / ∈ Σ, and Lemma 5.1 (i),we get[[ Y λ , X α ] , θX α ] = − [[ θX α , Y λ ] , X α ] − [[ X α , θX α ] , Y λ ] = 2 h α, λ i Y λ = | α | A α,λ Y λ . Thus, recalling the definition of the unit vector Y λ + α , we have [ Y λ + α , θX α ] = −| α | p − A α,λ Y λ .This finishes the proof. (cid:3) From now on, for each λ ∈ Σ + such that λ + α ∈ Σ and λ − α / ∈ Σ (e.g. for λ = β ), andfor a unit vector Y λ ∈ g λ , we define(28) Y λ + α := [ Y λ , X α ] | α | p − A α,λ ∈ g λ + α , which is a unit vector in g λ + α by means of Lemma 5.11. Note that Y λ + α depends on thechoice of Y λ , but we remove this dependence from the notation for simplicity. We can nowproceed with the calculation of the shape operator. Proposition 5.12.
Let Y λ ∈ g λ be a unit vector, for some λ ∈ Σ + \ { α } . Then, (29) S ξ Y λ = a | α | A α,λ Y λ − b Y λ , X α ] + b Y λ , θX α ] . In particular, if λ ∈ Σ + \ { α } is such that λ + α ∈ Σ but λ − α / ∈ Σ , then: S ξ Y λ = a | α | A α,λ Y λ − b | α | p − A α,λ Y λ + α . (30) S ξ Y λ + α = a | α | A α,λ + α Y λ + α + Y λ +2 α − b | α | p − A α,λ Y λ , (31) for some Y λ +2 α ∈ g λ +2 α . (If λ + 2 α / ∈ Σ then Y λ +2 α = 0 .)Proof. First, using Lemma 5.8 and Lemma 5.9 (i)-(iii), we get S ξ Y λ = −∇ Y λ ξ = −
12 ([ Y λ , ξ ] + [ θY λ , ξ ] − [ Y λ , θξ ]) n = −
12 [ Y λ , ξ ] + 12 [ Y λ , θξ ]= −
12 [ Y λ , aH α + bX α ] + 12 [ Y λ , − aH α + bθX α ] = a h α, λ i Y λ − b Y λ , X α ] + b Y λ , θX α ]= a | α | A α,λ Y λ − b Y λ , X α ] + b Y λ , θX α ] . This proves (29). Note that (30) follows directly from (29), taking into account (28) and λ − α / ∈ Σ. Finally, using (29) and Lemma 5.11 we get S ξ Y λ + α = a | α | A α,λ + α Y λ + α − b Y λ + α , X α ] + b Y λ + α , θX α ]= a | α | A α,λ + α Y λ + α + Y λ +2 α − b | α | p − A α,λ Y λ , where we put Y λ +2 α := − b [ Y λ + α , X α ] / ∈ g λ +2 α . This proves (31). (cid:3) Therefore, taking into account the definition of the endomorphism D in (22), Proposi-tion 5.10 and Proposition 5.12, we can state the following: Corollary 5.13.
Let ξ = aH α + bX α be a unit vector, where X α ∈ g α is a unit vector, α ∈ Π , and a , b ∈ R . Let λ ∈ Σ + \{ α } be such that λ + α ∈ Σ and λ − α / ∈ Σ . Let Y λ ∈ g λ ⊂ s be a unit vector. Then there exists Y λ +2 α ∈ g λ +2 α such that: DY λ = tr( S ξ ) (cid:18) a | α | A α,λ Y λ − b | α | p − A α,λ Y λ + α (cid:19) + cY λ ,DY λ + α = tr( S ξ ) (cid:18) a | α | A α,λ + α Y λ + α + Y λ +2 α − b | α | p − A α,λ Y λ (cid:19) + cY λ + α . Corollary 5.13 with λ = β gives us the expression of D | g β ⊕ g β + α . Thus, so far, we havecalculated D restricted to the subspaces a ⊖ R H α and g β ⊕ g β + α of s . Hence, calculating DU is the last step in order to determine the endomorphism D restricted to the subspace (23)of s . Recall that U = b | α | − H α − a | α | X α is a unit vector in s . Let us first introduce thefollowing auxiliary result. Lemma 5.14.
We have: ∇ X α ξ = | α | U, (i) ∇ U ξ = − a | α | U , (ii) ∇ U U = a | α | ξ , (iii) ∇ ξ U = − b | α | ξ , (iv) [ U, ξ ] = | α | X α , (v) ∇ ξ ξ = b | α | U . (vi) Proof.
In this proof, let Z ∈ a ⊕ n be arbitrary. Firstly, using (21), Lemma 5.1 (i) and (20),we have h∇ X α ξ, Z i = 14 h [ X α , aH α + bX α ] + [ θX α , aH α + bX α ] − [ X α , θ ( aH α + bX α )] , Z i B θ = 14 h− a [ H α , X α ] + b [ θX α , X α ] − a [ H α , X α ] − b [ X α , θX α ] , Z i B θ = − a | α | h X α , Z i B θ + b h H α , Z i B θ = − a | α | h X α , Z i + b h H α , Z i = h| α | U, Z i . This proves claim (i). Using Lemma 5.2 (iii) and claim (i) we get ∇ U ξ = b | α | − ∇ H α ξ − a | α |∇ X α ξ = − a | α | U, which proves claim (ii). Now, using again (21), Lemma 5.1 (i) and (20), we obtain h∇ X α U, Z i = 14 h [ X α , U ] + [ θX α , U ] − [ X α , θU ] , Z i B θ = 14 h− b | α | − [ H α , X α ] − a | α | [ θX α , X α ] − b | α | − [ H α , X α ] + a | α | [ X α , θX α ] , Z i B θ = − a | α |h H α , Z i B θ − b | α |h X α , Z i B θ = − a | α |h H α , Z i − b | α |h X α , Z i . Hence ∇ X α U = −| α | ξ , and then, using Lemma 5.2 (iii), we get ∇ U U = b | α | − ∇ H α U − a | α |∇ X α U = a | α | ξ, ∇ ξ U = a ∇ H α U + b ∇ X α U = − b | α | ξ, which proves (iii) and (iv). Now, taking into account that a | α | + b = 1 since ξ is a unitvector, we get[ U, ξ ] = [ b | α | − H α − a | α | X α , aH α + bX α ] = b | α | X α + a | α | X α = | α | X α , which proves assertion (v). Finally, claim (vi) follows directly from Lemma 5.2 (ii). (cid:3) Proposition 5.15. DU = (tr( S ξ ) a | α | + b | α | + c ) U .Proof. Using Lemma 5.14 several times, and the relation a | α | − − b , we get( R ξ + S ξ ) U = ∇ U ∇ ξ ξ − ∇ ξ ∇ U ξ − ∇ [ U,ξ ] ξ + ∇ ∇ U ξ ξ = b | α |∇ U U + a | α | ∇ ξ U − | α |∇ X α ξ − a | α | ∇ U ξ = ab | α | ξ − ab | α | ξ − | α | U + a | α | U = − b | α | U. Moreover, by Lemma 5.14 (ii) we have S ξ U = −∇ U ξ = a | α | U . Altogether: DU = (tr( S ξ ) S ξ − ( R ξ + S ξ ) + c Id) U = (tr( S ξ ) a | α | + b | α | + c ) U. (cid:3) Proposition 5.7, Corollary 5.13 and Proposition 5.15 provide an explicit description ofthe endomorphism D defined in (22) restricted to the subspace (23) of s . We are thereforein position to prove Theorem 5.6. Proof of Theorem 5.6.
As above in this section, we assume that M has rank greater thanone, s = ( a ⊕ n ) ⊖ R ξ is a Lie subalgebra of a ⊕ n with ξ = aH α + bX α of unit length, forsome α ∈ Π, some unit vector X α ∈ g α , and a , b ∈ R , and S is an algebraic Ricci soliton,that is, the endomorphism D given in (22) is a derivation of s . The result will be provedif we show that b = 0.The first step will be to show c = 0, where c is the scalar involved in (22). In order todo so, select a root λ ∈ Σ + \ { α } such that λ + α ∈ Σ and λ − α / ∈ Σ. Recall that such aroot exists since M is irreducible and of rank greater than one by assumption: it sufficesto choose a simple root λ connected to α in the Dynkin diagram of M . Now take a unitvector Y λ ∈ g λ ⊂ s . Since λ is not proportional to α and dim a ≥
2, there exists an element H ∈ a ⊖ R H α ⊂ s such that λ ( H ) = h H, H λ i 6 = 0. As D is assumed to be a derivation, then(32) D [ H, Y λ ] = [ DH, Y λ ] + [ H, DY λ ] . Note that D [ H, Y λ ] = λ ( H ) DY λ . Moreover, by Proposition 5.7, we have [ DH, Y λ ] = cλ ( H ) Y λ . Since by assumption h H, H α i = 0, then ( λ + α ) H = λ ( H ), and hence byCorollary 5.13 we get [ H, DY λ ] = λ ( H ) DY λ . Altogether, (32) now becomes λ ( H ) DY λ = cλ ( H ) Y λ + λ ( H ) DY λ . Since λ ( H ) = 0 and Y λ is a non-zero vector, we get c = 0, as desired.In order to conclude that b = 0, we will examine the relation D [ U, Y λ ] = [ DU, Y λ ] +[ U, DY λ ], which holds because D is a derivation. In particular, we have(33) h D [ U, Y λ ] , Y λ i = h [ DU, Y λ ] , Y λ i + h [ U, DY λ ] , Y λ i . Recalling from (28) the definition of Y λ + α , we get[ U, Y λ ] = (cid:2) b | α | − H α − a | α | X α , Y λ (cid:3) = b | α | A α,λ Y λ + a p − A α,λ | α | Y λ + α . (34) Now, using (34) and Corollary 5.13 along with c = 0, we obtain(35) h D [ U, Y λ ] , Y λ i = b | α | A α,λ h DY λ , Y λ i + a p − A α,λ | α | h DY λ + α , Y λ i = tr( S ξ ) ab | α | A α,λ + tr( S ξ ) ab | α | A α,λ . From Proposition 5.15 and c = 0 we get DU = ( a tr( S ξ ) + b ) | α | U , and then by (34)we have(36) h [ DU, Y λ ] , Y λ i = ( a tr( S ξ ) + b ) | α | h [ U, Y λ ] , Y λ i = b a tr( S ξ ) + b ) | α | A α,λ . Corollary 5.13, (34), and the fact that [
U, Y λ + α ] ∈ g λ + α ⊕ g λ +2 α yield(37) h [ U, DY λ ] , Y λ i = a S ξ ) | α | A α,λ h [ U, Y λ ] , Y λ i − b S ξ ) p − A α,λ | α |h [ U, Y λ + α ] , Y λ i = tr( S ξ ) ab | α | A α,λ . Finally, inserting (35), (36) and (37) into (33), we get b | α | A α,λ = 0. As λ − α / ∈ Σ and λ + α ∈ Σ, we have A α,λ = 0 (see [19, Proposition 2.48 (g)]). Since of course | α | 6 = 0 as α ∈ Σ, we conclude that b = 0, as we wanted to show. (cid:3) We can now put several results together to conclude the proof of Theorem A.
Proof of Theorem A.
Let S be a codimension one subgroup of the solvable part AN ofan Iwasawa decomposition of a symmetric space of non-compact type M . Let us write s = ( a ⊕ n ) ⊖ R ξ for the Lie algebra of S . First note that, since a ⊕ n is completely solvable,the same is true for the Lie subalgebra s = ( a ⊕ n ) ⊖ R ξ . Hence, by [22], the connected,simply connected Lie subgroup S of the Iwasawa group AN with Lie algebra s is a Riccisoliton if and only if it is an algebraic Ricci soliton.By Lemma 5.3, if S is an algebraic Ricci soliton, then either ξ ∈ a , or ξ / ∈ a and ξ ∈ R H α ⊕ g α for some α ∈ Π. In the first case S is a connected Lie subgroup of AN containing N ; and, conversely, any connected Lie subgroup of AN containing N is asolvsoliton by virtue of Proposition 5.5.In the second case, Theorem 5.6 implies that M has rank one. Therefore, M is a hyper-bolic space and, hence, homothetic to a Damek-Ricci space or to a real hyperbolic space R H n , as recalled in § AN of an Iwasawa decom-position associated with M has the structure of a Damek-Ricci space, up to homothety.Then Theorem C implies that S is an algebraic Ricci soliton if and only if S is eitherthe subgroup N of AN (that is, a horosphere in the rank one symmetric space M ) or aLohnherr hypersurface W in a complex hyperbolic plane C H . In the second subcase, if M is a real hyperbolic space R H n , the isometric action of S on R H n is of cohomogeneityone and without singular orbits (as S is a codimension one subgroup of AN , which actssimply transitively on R H n ). Then any of the orbits of such action is isoparametric and,by the theory of isoparametric hypersurfaces in space forms (see for example [10, § by the Gauss equation of submanifold geometry, such orbits always have constant sectionalcurvature and are therefore Einstein. (cid:3) References [1] R. M. Arroyo, R. Lafuente: Homogeneous Ricci solitons in low dimensions,
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Ann. Glob. Anal. Geom. (2011), 291–309. Department of Mathematics, Universidade de Santiago de Compostela, Spain.
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