Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups
aa r X i v : . [ m a t h . DG ] S e p Concerning the existence of Einstein and Ricci soliton metrics onsolvable Lie groups
M.Jablonski
Abstract
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics ofinterest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that theexistence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how onemay determine the existence of such a metric by analyzing algebraic properties of the Lie algebra inquestion and infinitesimal deformations of any initial metric.Our second main result concerns the isometry groups of such distinguished metrics. Among thecompletely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits sucha metric, we show that the isometry group of this special metric is maximal among all isometry groupsof left-invariant metrics. We finish with a similar result for locally left-invariant metrics on compactnilmanifolds.
Our primary interest in this work is (left-invariant) Einstein metrics on non-compact Lie groups. All knownexamples of such metrics occur on solvable Lie groups. In fact, all known examples of non-compact homo-geneous Einstein metrics are isometric to solvable Lie groups with left-invariant metrics; this is the contentof the well-known Alekseevskii conjecture which has been verified in dimensions 4 and 5 [Jen69, Nik05]. Werestrict ourselves to this class of Lie groups and ask when such a group admits an Einstein metric.The answer in the compact setting is well-known. If a compact group G admits an Einstein metric, theneither(i) G is a torus (zero scalar curvature) or(ii) G is a compact semi-simple Lie group (positive scalar curvature).The first case of Ricci flat follows from a general result of Alekseevskii-Kimelfeld where it is shown that anyhomogeneous Ricci flat space is actually flat [AK75]. For the positive scalar curvature case, such metricsare characterized as critical points of the total scalar curvature functional [Jen71]. On compact semi-simplegroups, Einstein metrics are not unique and most groups admit at least 2 such metrics (this is in sharpcontrast to the solvable setting, where Einstein metrics are unique up to isometry and scaling).We observe that the existence of an Einstein metric on a compact Lie group is completely determinedby the underlying Lie algebra. The flat case corresponds to abelian Lie algebras. The positive case ischaracterized as follows. Let g denote the Lie algebra of G , then G is compact semi-simple if and only if theKilling form B ( X, Y ) = tr ( ad X ◦ ad Y ), with X, Y ∈ g , is negative definite.Our work is motivated by, and seeks to answer, the following questions. Question 1.1.
Given a solvable Lie group, how can one determine if it admits an Einstein or solsolitonmetric?
Question 1.2.
If a solvable Lie group is known to admit an Einstein or solsoliton metric, how does onefind it?
1s the curvature tensors of left-invariant metrics are left-invariant, the above questions reduce to studyinginner products on a given Lie algebra. More precisely, consider a solvable Lie algebra g with correspondingLie group G . Let h· , ·i be an inner product on g with corresponding left-invariant metric on G . The Riccicurvature of ( G, h· , ·i ) is completely determined by its values on g (by left-invariance) and is given by theformula ric ( X, X ) = − X i | [ X, X i ] | − X i h [ X, [ X, X i ]] , X i i + 14 X i,j h [ X i , X j ] , X i − h [ Z, X ] , X i where { X i } is an orthonormal basis of g and Z ∈ g is the unique vector satisfying h Z, X i = tr ( ad X ). Observethat the Ricci curvature is completely determined by the Lie bracket and inner product on g . Denoting the(1 , Ric , Question 1.1 may be rephrased as follows.
Question.
When does there exist an inner product h· , ·i on g , such that Ric = cId + D (1.1) for some c ∈ R and some D ∈ Der ( g ) ? Here
Der ( g ) denotes the algebra of derivations of g . Terminology: when D = 0, the metric is called anEinstein metric; when D = 0, the metric is called a solsoliton. Solsolitons are algebraic examples of Riccisolitons, see Section 2.Our main result shows that a definite answer to Question 1.1 can be obtained by analyzing only localdata: algebraic information about the underlying Lie algebra and infinitesimal deformations of any metric;see Section 10 for complete details and the procedure referenced by the following theorem. Theorem 10.1.
Let G be a solvable Lie group with Lie algebra g . The existence of a left-invariant Einsteinmetric on G can be determined by analyzing the following: 1) adjoint action of g on itself, 2) the commutatorsubalgebra n = [ g , g ] , and 3) infinitesimal deformations of any initial metric on n . Remark.
The existence of an Einstein metric on a solvable Lie group is now a local question. Similarly,one can formulate the question of existence of a solsoliton in terms of local data.
In general, the existence of an Einstein or Ricci soliton metric is not a local question. It might appearat first glance that the existence of left-invariant Einstein metrics on Lie groups is a local question since theverification of Equation 1.1 uses only the inner product and Lie bracket on g . However, asking if a Lie groupadmits such a metric amounts to asking if there exists a zero of the function || Ric g − sc ( g ) n Id || on the openset of inner products. It is not clear if this is a local question for non-solvable Lie groups; e.g., there doesnot exist a solution when the Lie algebra is sl R .In the setting of compact homogeneous spaces G/H , the Einstein question has received a great deal ofattention and there are some partial results on the existence of such metrics. For example, if G is a compactsemi-simple group and H is a maximal connected subgroup of G , then G/H admits a G -invariant Einsteinmetric [WZ86]. However, there exist many examples of homogeneous spaces G/H which don’t admit G -invariant Einstein metrics. Presently, there are not any general, local conditions that guarantee/exclude theexistence of such metrics on compact homogeneous spaces; see [BWZ04] for the current state of research.Our work builds on the strong structural results of Heber [Heb98] and Lauret [Lau07]. These works takethe first step in reducing the problem on the solvable group to a smaller solvable group, a one dimensionalextension of a nilpotent group. This smaller solvable Lie group admits an Einstein metric if and only ifits nilradical admits a so-called nilsoliton metric and the underlying Lie algebra is the extension of thenilradical by a so-called pre-Einstein derivation. Reducing the problem to analyzing the nilradical is analgebra problem.To study the nilradical we build on the work of Nikolayevsky [Nik08a]. Using a combination of measuringalgebraic information and infinitesimal deformations of metrics on the nilradical, we translate the Einstein2roblem into a local problem. (While we could skip this analysis on the nilradical and couple our techniquesdirectly with the work of Heber [Heb98, Section 6], we present our results in the given framework as thesemethods extend directly to solsoliton and nilsoliton metrics. See Section 9 for more details.)Our second main result is an algebraic decomposition theorem for solvable groups admitting Einsteinmetrics. If one were to classify the solvable groups admitting Einstein or solsoliton metrics, one would wantto construct such groups from basic building blocks. The question of existence of an Einstein or solsolitonmetric can be reduced to the case that the underlying Lie algebra is indecomposable. Theorem 4.8
Let G be a solvable Lie group whose Lie algebra g = g + g is a direct sum of ideals. Then G admits a non-flat solsoliton, resp. flat, metric if and only if both G and G admit non-flat solsoliton, resp.flat, metrics. Corollary 4.9
Let G be a solvable Lie group whose Lie algebra g = g + g is a direct sum of ideals. Then G admits an Einstein metric if and only if both G and G admit Einstein metrics of the same sign. Remark.
A similar decomposition result has appeared for nilsolitons and nilpotent Lie groups, see [Nik08a]and [Jab08a]. To our knowledge, the above algebraic decomposition theorem is the first of its kind forhomogeneous Einstein spaces. It would be interesting to know if there is a similar theorem in the compactsetting; there are some partial results of B¨ohm in this direction [B¨oh04, Theorem B].
In addition to providing a local formulation of the existence of such a metric on a solvable Lie group, wedemonstrate how to recover such metrics by following two natural curves of metrics (see Proposition 5.4).Using these curves, we demonstrate that solsolitons (when they exist) are the most symmetric metric oncompletely solvable unimodular Lie groups (this class includes nilpotent Lie groups).
Theorem 5.8
Let S be a completely solvable unimodular Lie group that admits a solsoliton metric. Let g be any left-invariant metric. Then there exists a left-invariant soliton metric g ′ such that Isom ( S, g ) ⊂ Isom ( S, g ′ ) , as groups. This result is extended to compact nilmanifolds with local nilsoliton metrics in Theorem 6.3.
Table of Contents.
Section 2 reviews information on Lie groups with left-invariant metrics. Section 3discusses the space of Lie brackets, moment maps, and distinguished orbits. Section 4 compares the existenceof solsoliton metrics and distinguished orbits. Section 5 studies the bracket flow with applications to findingsolsoliton metrics and comparisons of isometry groups. Section 6 states results on compact nilmanifolds.Section 7 discusses the stratification of the space of Lie brackets. Section 8 covers pre-Einstein derivationsand the previous work of Nikolayevsky. Sections 9 and 10 show the existence of nilsoliton and Einsteinmetrics are intrinsic to the underlying Lie algebra, respectively.
A Lie group G is called a Riemannian Lie group if it is endowed with a left-invariant metric. The followingquestion motivates much of our work.
Question 2.1.
Among left-invariant metrics on a given Lie group, is there a canonical or preferred choiceof metric?
Special metrics are often characterized as those having good curvature properties or as solutions tosome extremal problem. For example, we are interested in metrics satisfying one or several of the followingconditions 3. Nice curvature properties or curvature tensor2. Critical points of a Riemannian functional on the set of metrics3. More generally, fixed points of a dynamical system4. Large group of isometriesClassically, spaces with constant sectional and Ricci curvature have been investigated as preferred metricson a manifold; the later are known as
Einstein metrics . We are interested in left-invariant Einstein metricswhen they exist; however, many of our Lie groups are not able to admit left-invariant Einstein metrics. Forexample, any non-abelian nilpotent Lie group cannot admit an Einstein metric [Jen69, Mil76] and manysolvable Lie groups do not admit Einstein metrics (cf. Theorem 5.11).Given that many of our Lie groups will not be able to admit an Einstein metric, we explore alternatemetrics in search of one which is ‘distinguished’ or preferred in some way. One natural generalization of theEinstein metric is the so-called
Ricci soliton metric . A metric g is called a Ricci soliton if there exists acomplete vector field X ∈ X ( G ) and constant c ∈ R such that ric g = cg + L X g where L X is the Lie derivative generated by X . These metrics arise as special solutions to the Ricci flowwhich are of the form g ( t ) = c ( t ) ϕ ∗ ( t ) g where c ( t ) is a real-valued function and ϕ ( t ) is the 1-parameter groupof diffeomorphisms that generates X ∈ X ( G ). Hence, Ricci solitons can be viewed as generalized fixed pointsof the Ricci flow on the space of metrics modulo diffeomorphisms; see [CK04] for an introduction to Ricciflow and Ricci solitons.On a Riemannian Lie group G there is a very natural kind of Ricci soliton, which we call an algebraicRicci soliton, or algebraic soliton for short. A left-invariant metric g is called an algebraic soliton if the Riccitensor (evaluated at the identity e ∈ G ) is of the form Ric = cId + D for some D ∈ Der ( g ). In the above equation, we have written the Ricci (1 , Ric instead of the (2 , ric as in the definition of Ricci soliton above; this is done for ease of presentation. The derivation D generates the 1-parameter family of automorphisms ϕ ( t ) = exp ( tD ) which is the corresponding family ofdiffeomorphisms from the definition of Ricci soliton. Remark.
1) By left-invariance of the metric on G , it suffices to only prescribed the value of Ric at e ∈ G .2) Presently, the only known examples of Ricci solitons on Riemannian Lie groups are algebraic solitons andthese are only known to exist on solvable Lie groups. When G is a nilpotent group, such a metric is often called a nilsoliton in the literature and if G is solvable,such a metric has been called a solsoliton . Nilpotent Lie groups
As nilpotent Lie groups cannot admit Einstein metrics, we will search for left-invariant Ricci soliton metricson such spaces. These metrics satisfy several of the criteria listed above for being a preferred metric. Beforestating the next theorem, we need some notation. Given a nilpotent Lie group G , denote by M G the set ofleft G -invariant metrics on G (i.e. inner products on g ). Recall that Ric h is left G -invariant for h ∈ M G . Theorem 2.2. [Lau01a, Jab10]i. (Algebro-analytic characterization)Every left-invariant Ricci soliton on a nilpotent Lie group G is an algebraic soliton, that is, Ric = cId + D for some D ∈ Der ( g ) . i. (Variational characterization)A metric g ∈ M G is a nilsoliton if and only if g is a critical point of the functional F ( h ) = tr Ric h sc ( h ) on M G , where sc ( h ) denotes the (constant) scalar curvature of ( G, h ) . (This functional makes senseas a real-valued function by left-invariance of Ric .)iii. (Uniqueness)If a nilsoliton exists on G , then it is unique (up to isometry) after prescribing the scalar curvature. Remark 2.3.
The above results are primarily due to Lauret [Lau01a]. Part ii. was originally proven wherethe functional considered was on the space of Lie brackets, this result has been extended to the setting ofleft-invariant Riemannian metrics in [Jab10] where new convergence results are also obtained.There is a small gap in the original proof of Part i. above. In that work, it is shown that if the metric isa soliton, then there exists a derivation D ∈ Der ( g ) such that Ric = cId + ( D + D t ) . The gap is fixableby showing that there exists such D satisfying D t ∈ Der ( g ) ; this will appear in a future work of that author[Lau]. Observe that g ∈ M G is a nilsoliton if and only if g is a critical point of the functional F ( h ) = tr Ric h along the set { h ∈ M G | sc ( h ) = sc ( g ) } . A similar functional can be studied on compact nilmanifolds, seeSection 6.In the sequel, we will see that nilsolitons have maximal isometry groups among all left-invariant metrics.More precisely we have the following result, see Corollary 5.9. Corollary
Let G be a nilpotent Lie group which admits a nilsoliton metric. Let g ∈ M G be any left-invariantmetric on G . Then there exists a nilsoliton g ′ ∈ M G on G such that Isom ( g ) ⊂ Isom ( g ′ ) . Remark 2.4.
In this way, we see that nilsolitons are the most symmetric left-invariant metric on a nilpotentLie group. However, there are other non-soliton metrics which can have the same isometry group. Itwould be interesting to know which other geometric properties these highly symmetric nilmanifolds share withnilsolitons.
The property of being an Einstein nilradical is intrinsic to the underlying Lie algebra. Moreover, one canverify this via an algorithm, see Theorem 9.1 and Section 9.
Theorem
Let N be a nilpotent Lie group with Lie algebra n . The existence of a nilsoliton metric ona nilpotent Lie group N can be determined by analyzing the derivation algebra Der ( n ) and infinitesimaldeformations of any initial metric on n . Solvable Lie groups
The analysis of Riemannian solvable Lie groups splits into two distinct sets: unimodular and non-unimodulargroups. A Lie group G is called unimodular if | det ( Ad ( g )) | = 1 for all g ∈ G . Notice, in particular, that tr ad ( X ) = 0 for X ∈ g when G is unimodular. If G is not unimodular it is called non-unimodular .Within both these classes of solvable Lie groups, we are interested in those which are completely solvable.A solvable group G is called completely solvable if ad ( X ) : g → g has only real eigenvalues, for all X ∈ g .Observe that nilpotent Lie groups are unimodular completely solvable, with eigenvalues all zero.As (non-abelian) solvable Lie groups are non-compact, any left-invariant Einstein metric on such a groupmust have scalar curvature less than or equal to zero by Bonnet-Myers theorem. We recall the followingresult (cf. [DM82, AK75]). 5 heorem 2.5 (Alekseevskii-Kimel’fel’d, Dotti) . Let G be a solvable Lie group with left-invariant Einsteinmetric.1. The scalar curvature is negative if and only if G is non-unimodular, and2. The scalar curvature is zero if and only if G is unimodular.In the case of Ricci flat, the metric is actually flat (i.e., constant zero sectional curvature). In [AK75] it is actually shown that any homogeneous Ricci flat space must be flat. Lie groups with flatleft-invariant metrics are necessarily unimodular solvable and have been classified [Mil76].
Theorem 2.6 (Milnor) . A Riemannian Lie group G is flat if and only if its Lie algebra g (with innerproduct) splits as an orthogonal direct sum g = a ⊕ n where n is an abelian ideal (the nilradical) and a is anabelian Lie algebra such that ad X is skew-symmetric for X ∈ a . Such G is necessarily solvable. Given the simple algebraic structure of these solvable Lie groups, one may classify the solvable Lie groupsadmitting flat metrics. These are the groups whose Lie algebras are constructed as follows (cf. Theorem5.11 where the negative Einstein case is considered).
Proposition 2.7.
Let n be an abelian Lie algebra and a ⊂ Der ( n ) an abelian, reductive subalgebra ofderivations, all of whose elements have purely imaginary eigenvalues. If G is a (solvable) Lie group whoseLie algebra is the semi-direct product a ⊕ n , then G admits a flat metric. Conversely, every such solvablegroup arises this way.Proof. Picking a basis of n , we may identify it with R n . Via this identification, the abelian, reductive algebra a ⊂ gl ( n, R ).It is well-known that there exists an inner product on R n such that a is stable under the transposeoperation, see [Mos55]. As the eigenvalues of every element in a are purely imaginary, we see that a consistsof skew-symmetric derivations. Now Milnor’s theorem above applies.As the unimodular solvable Lie groups admitting Einstein metrics are understood, our attention is dedi-cated to analyzing the non-unimodular solvable groups admitting Einstein metrics and both kinds of solvablegroups which admit solsoliton metrics. As in the case of nilpotent Lie groups, solsolitons (including Einsteinmetrics) have several rigid properties. The following results may be found in [Lau10]. Theorem 2.8 (Lauret) . Let ( G, g ) be a solvable Riemannian Lie group with metric Lie algebra ( g , g ) . Let n be the nilradical of g (with induced metric) and a = n ⊥ , so that g = a ⊕ n .i. (Structural results and the standard property)The Riemannian Lie group G is a solsoliton (i.e. Ric = cId + D ) if and only ifa) ( n , g ) (with the induced metric) is a nilsolitonb) a = n ⊥ is abelianc) ( ad A ) t ∈ Der ( g ) (or equivalently, [ ad A, ( ad A ) t ] = 0 ) for all A ∈ a .d) g ( A, A ) = − c tr S ( ad A ) for all A ∈ a , where S ( ad A ) = ( ad A + ( ad A ) t ) ii. (Solsolitons are not shrinkers)The constant c satisfying Ric = cId + D satisfies c ≤ . Moreover, if c = 0 then D = 0 and the metricis flat (cf. Theorem 2.6). This says solsolitons are either so-called steady or expanding Ricci solitons(as opposed to shrinking solitons).iii. (Uniqueness)If a solvable Lie group admits a solsoliton, then it is unique (up to isometry) after prescribing the scalarcurvature. Observe that Part i.d) implies that the eigenvalues of ad A are not all purely imaginary, for any A ∈ a .6 emark 2.9.
1) There does exist a variational characterization for Einstein solvmanifolds with codimen-sion 1 nilradical which realizes these spaces as critical points of a ‘modified scalar curvature function’ (see[Lau01b]). 2) As in the case of flat Einstein metrics, we have a characterization of solvable Lie groups whichadmit Einstein and solsoliton metrics, see Theorem 5.11 and Corollary 5.12.
Let G be a solvable unimodular Lie group. If G admits a non-trivial solsoliton, it cannot admit a (flat)Einstein metric, and conversely, if G admits a (flat) Einstein metric, it cannot admit a non-trivial solsoliton.In this way, solsolitons are a preferred metric on unimodular solvable groups that cannot admit (flat) Einsteinmetrics. This preference is defended by the following, see Theorem 5.8. Theorem.
Let G be unimodular completely solvable Lie group which admits a solsoliton metric. Given anyleft-invariant metric g , there exists a solsoliton metric g ′ such that Isom ( g ) ⊂ Isom ( g ′ ) . Remark 2.10.
Presently, we do not have such a theorem when the group is not completely solvable or fornonunimodular solvable Lie groups. Our techniques do allow one to embed a large portion of the isometrygroup of any metric into the Einstein or solsoliton metric, however they do not allow one to embed the entireisometry group. This question will be addressed in future work.
Isometry groups
The group of isometries of a Riemannian solvable Lie group is particularly simple when the group in questionis a completely solvable unimodular group. The following is Theorem 4.3 of [GW88].
Theorem 2.11 (Gordon-Wilson) . Let G be a completely solvable unimodular Lie group with left-invariantmetric g . The full isometry group is a semi-direct product Isom ( G, g ) = K ⋉ G where K ⊂ Aut ( G ) is the isotropy subgroup of Isom ( G, g ) preserving the identity e ∈ G . Under the naturalidentification Aut ( G ) ≃ Aut ( g ) we have K ≃ Aut ( g ) ∩ O ( g ) , where O ( g ) is the orthogonal group of the innerproduct g on g . Observe that this theorem covers the case of any Riemannian nilpotent Lie group.
Definition 2.12.
Let ( G, g ) be a Riemannian Lie group. The group G ⋊ ( Isom ( g ) ∩ Aut ( G )) is a subgroupof isometries which we call the algebraic isometry group. The above theorem says that the algebraic isometry group of a completely unimodular solvable group isthe whole isometry group. For non-unimodular solvable groups, it is well-known that the full isometry groupis significantly larger than its algebraic isometry group [GW88].
A Lie group G with a left-invariant metric h , i gives rise to a metric Lie algebra { g , h , i} , where g is the Liealgebra of G and the inner product on g is the restriction of the left-invariant metric to T e G ≃ g . Conversely,a metric Lie algebra gives a left-invariant metric on any Lie group with said Lie algebra. We are primarilyinterested in simply-connected Lie groups.We say that two metric Lie algebras { g , h , i } and { g , h , i } are isomorphic if there exists a Liealgebra isomorphism φ : g → g such that h , i = φ ∗ h , i . Such an isomorphism lifts to give an isometrybetween the simply-connected Riemannian Lie groups { G , h , i} → { G , h , i} . In general, most isometriesdo not arise this way, however, for nilpotent and some solvable groups, this is how all isometries arise (seeTheorem 2.11).To obtain good information on Riemannian Lie groups, we study metric Lie algebras by considering ametric Lie algebra as a collection of three objects: a vector space R n , a Lie bracket [ · , · ] and an inner product7 · , ·i . We use what is becoming a standard technique and convert our questions into a frame work that canexploit deep theorems from Geometric Invariant Theory: Instead of fixing a Lie algebra and varying an innerproduct on it, we choose to fix an inner product and vary the underlying Lie algebra structure (within thesame isomorphism class).For g ∈ GL ( n, R ), we may consider a different (and isomorphic) Lie bracket g ∗ [ · , · ] = g [ g − · , g − · ] andthe inner product g ∗ h· , ·i = h g − · , g − ·i . The following are isomorphic metric Lie algebras { R n , g ∗ [ · , · ] , h· , ·i} ≃ { R n , [ · , · ] , ( g − ) ∗ h· , ·i} via the isomorphism g − : R n → R n .We now fix an inner product (the usual one) on R n and study the collection g ∗ [ · , · ], g ∈ GL ( n, R ). It ishelpful to study not just this collection of isomorphic Lie algebras on R n , but instead to study all Lie algebrastructures on R n . Consider the vector space V = ∧ ( R n ) ∗ ⊗ R n , the space of anti-symmetric, bilinear mapsfrom R n × R n → R n . This vector space is endowed with a natural GL ( n, R ) action:( g ∗ [ · , · ])( v, w ) = g [ g − v, g − w ]for g ∈ GL ( n, R ), v, w ∈ R n . Via differentiation, we also have an action of gl ( n, R ) on V . Given X ∈ gl ( n, R )and v, w ∈ R n , we have ( X · [ · , · ])( v, w ) = X [ v, w ] − [ Xv, w ] − [ v, Xw ].The points of V can be thought of as anti-symmetric algebra structures on R n , and two algebra structuresare isomorphic if and only if they lie in the same GL ( n, R )-orbit. Any Lie bracket [ · , · ] on R n can be realizedas a point in V and the subset V = { µ ∈ V | µ satisfies the Jacobi identity } is a variety in V whose points are the Lie brackets on R n . Additionally, we will restrict our attention tosome interesting subsets of V : let N denote the Lie brackets which are nilpotent, S denote the Lie bracketswhich are solvable, and CS denote the Lie brackets which are completely solvable (cf. Section 2). We havethe following containments N ⊂ CS ⊂ S
These subsets are all closed in V . We will often abuse language and refer to µ ∈ V as a Lie algebra, whenwe really mean the pair { R n , µ } .Given a Lie bracket µ ∈ V , we will denote by s µ the metric Lie algebra { R n , [ · , · ] , h· , ·i} ; the correspondingsimply connected Lie group with left-invariant metric will be denoted by S µ . Remark 3.1.
For k ∈ O ( n, R ) , the groups S µ and S k · µ are isometric. However, in general, one often has S µ and S λ which are isometric but λ O ( n, R ) · µ . As it will be of interest later, we point out that the stabilizers of the actions of GL n R and gl n R haverelevant meaning: ( GL n R ) µ = Aut ( µ ) and ( gl n R ) µ = Der ( µ ). The moment map and geometry of orbits
The geometry of GL ( n, R )-orbits in V is intimately connected to algebraic properties of the Lie group S µ associated to µ and geometric structures that the group can admit. For example, consider the induced actionof SL ( n, R ) on V . For µ ∈ V , SL ( n, R ) · µ is closed in V if and only if µ is a semi-simple Lie algebra, see[Lau03].When µ is nilpotent, it is known that so-called distinguished orbits (which are generalizations of closedorbits, see Definition 3.2) correspond precisely to nilpotent Lie groups which admit left-invariant Ricci solitonmetrics, see Theorem 3.3. In the sequel, we show that distinguished orbits also play an role in the study ofsolvable Lie groups and solsolitons. 8efore defining distinguished orbits, we must define the moment map of a representation of a reductivegroup. The moment map defined here works for non-compact groups and is a natural extension of the usualone defined for compact groups, cf. [EJ09].The group GL ( n, R ) is endowed with the Cartan involution θ ( g ) = ( g t ) − , where ∗ t denotes the transposeoperation. By differentiating, we have an involution on gl ( n, R ) as well, which we denote by the same symbol: θ ( X ) = − X t .These involutions gives rise to so-called Cartan decompositions GL ( n, R ) = KP gl ( n, R ) = k ⊕ p where K = O ( n ) = { g ∈ GL ( n, R ) | θ ( g ) = g } , P = { g ∈ GL ( n, R ) | θ ( g ) = g − } , k = LieK = so ( n ) = { X ∈ gl ( n, R ) | θ ( X ) = X } , and p = symm ( n ) = { X ∈ gl ( n, R ) | θ ( X ) = − X } . Here symm ( n ) denotes thesymmetric n × n matrices. Additionally, P = exp ( p ), where exp is the Lie group exponential.Let G be a real algebraic reductive subgroup of GL ( n, R ) which is θ -stable. For such groups, we obtaina Cartan decomposition G = K G P G where K G = K ∩ G = G θ = { g ∈ G | θ ( g ) = g } is a maximal compactsubgroup of G and P G = P ∩ G = { g ∈ G | θ ( g ) = g − } . Similarly, the Lie algebra g = Lie G has a Cartandecomposition g = k G ⊕ p G . Often we will drop the subscript G when the group is understood.The (usual) inner product on R n extends of an O ( n )-invariant inner product h , i on the vector space V = ∧ ( R n ) ∗ ⊗ R n as follows h λ, µ i = X i An orbit G · v ⊂ V is called distinguished if it contains a critical point of the function F = || m || . v ∈ V is a critical point of F if and only if π ( m ( v ))( v ) = rv for some r ∈ R . It is a factthat any closed orbit is distinguished with critical value 0 and so these orbits are a natural generalization ofclosed orbits, see [Jab08b]. The following theorem motivates a deeper study of || m || on N ⊂ V . Theorem 3.3 (Lauret) . Let N µ denote the simply connected nilpotent Lie group with left-invariant metricwhose Lie algebra n µ (with inner product) corresponds to the point µ ∈ N . Then N µ is a nilsoliton if andonly if µ is a critical point of F ( v ) = || m || ( v ) . Equivalently, N µ is an Einstein nilradical if and only if theorbit GL n R · µ is distinguished. In this way, we convert our questions of left-invariant metrics on Lie groups into questions about thegeometry of orbits in the space V . By analyzing the geometry of orbits, we obtain our algorithm thatdetermines when a given nilpotent Lie group is an Einstein nilradical, see Section 9.The above theorem can be found in [Lau08]. The last equivalence is not stated using the label ofdistinguished orbit but is stated using the idea. In Section 10 of [Lau08] there are several open questions ofinterest which are presented. We state Question Question 3.4. Consider the function F : V → R defined by F ( v ) = || m ( v ) || where m is the moment map,as above. Define µ t to be the integral curve of − grad F starting at µ on the sphere of radius 2. Is µ ∞ (thelimit point along the integral curve) contained in the orbit GL n R · µ if N µ is an Einstein nilradical? In Theorem 4.2 we obtain an affirmative answer to this question. This result first appeared more gen-erally in [Jab08b], where this was shown to be true for distinguished orbits in any real reductive algebraicrepresentation. Remark 3.5. Geometrically, the moment map can be understood as follows. When µ is a nilpotent Liealgebra, m ( µ ) = 4 Ric ( N µ ) . Moregenerally, if µ is any Lie algebra with corresponding Lie group S µ , then m ( µ ) = 4 R where R is the tensor appearing in the formula Ric = R − B − S ( ad H ) here Ric is the Ricci tensor of S µ , B is the Killing form of the Lie algebra µ and S ( ad H ) = ( ad H + ad H t ) ,where ad H is a mean curvature vector. See [Bes08, Corollary 7.38] and [Lau10, Section 4] for more details. Soliton metrics on nilpotent Lie groups In the above section, we stated the relationship between nilsoliton metrics and critical points of the function F = || m || : they are precisely the same thing, see Theorem 3.3. As such, we are motivated to study thenegative gradient flow of F . Definition 4.1 (The bracket flow) . Let µ t ⊂ V denote the negative gradient flow of F starting at µ ∈ V . In Proposition 5.2, it will be seen that the limit of this flow is unique. We denote this limit point by µ ∞ . Theorem 4.2. Let N µ be an Einstein nilradical. Let µ ∞ denote the limit point of the negative gradientflow of the function F starting at µ . Then µ ∞ is contained in the orbit GL n R · µ ; i.e. N µ and N µ ∞ areisomorphic Lie groups. A more general result of this type is true for distinguished orbits and is useful for studying the geometryof solvable Lie groups, see Section 5. In the setting of nilpotent Lie algebras, the proof can be shorteneddramatically by employing the stratification results of Lauret (see Theorem 7.1). In addition, we obtainsome new and interesting geometric results on isometry and automorphism groups of nilpotent Lie groupsusing the techniques from this proof, see Corollary 5.9 and Proposition 7.6.10he above relationship between Einstein nilradicals and distinguished orbits has been studied extensivelyin the literature, see e.g. [Lau01a, Wil03, Pay10, LW07, Ebe08, Jab08b, Nik08a, Nik08b, Nik08c, Nik08d,Wil10, Jab09]. Motivated by this, we explore the relationship between distinguished orbits and solitonmetrics on solvable groups. Definition 4.3. A Riemannian Lie group S µ is said to have a distinguished metric if µ is a critical pointof F = || m || for the action of GL ( n, R ) on V (defined above). For a geometric interpretation of the moment map, see Remark 3.5. Einstein and soliton metrics on solvable Lie groups Proposition 4.4. If S µ is a solvable group admitting an Einstein or solsoliton metric, then GL ( n, R ) · µ isa distinguished orbit.Proof. To prove this, one consults the work [Lau03] where complex Lie algebras are studied. All the results ofthat paper remain true for real Lie algebras with the Hermitian transpose replaced with the usual transpose.For a detailed proof of this fact, see [Jab08b]. We warn the reader that the moment map defined there is amultiple of the moment map defined here. If n denotes the moment map from [Lau03] and m denotes themoment map used in this work, then n = 2 m . Our choice of moment map m is consistent with [Lau08, Lau10]Case 1: S µ admits a flat metric. If µ corresponds to the flat metric, then µ is also a critical point of F = || m || , see [Lau03, Theorem 4.7].Case 2: S µ admits a non-flat solsoliton. We only prove this in the case that the nilradical of s µ isnon-abelian. The abelian case is similar and we leave it to the diligent reader.The proof of this case is just a careful comparison of [Lau10, Theorem 4.8] with [Lau03, Theorem 4.7].The soliton metric and the distinguished metric differ only in their values on a × a , where a = n ⊥ . If thenilradical (which is a nilsoliton in either case) satisfies Ric n = cId + D , for some D ∈ Der ( n ), and has sc = − / 4, then the solsoliton metric on a is h A, A i = − c tr S ( ad A ) where S ( ad A ) is the symmetric part of ad A , while the distinguished metric on a is hh A, A ii = 12 · − c tr ( ad A ( ad A ) t )In [Lau03], a is viewed as a subset of Der ( n ) with A ≃ ad A . Remark 4.5. Observe that when S µ is completely solvable, a stronger statement is true. In this case, hh , ii = 12 h , i on a × a and the algebraic isometry groups (cf. Definition 2.12) are equal: Aut ( µ ) ∩ O ( hh , ii ) = Aut ( µ ) ∩ O ( h , i ) . Theorem 4.6. Let S µ be a completely solvable group. Then S µ admits a solsoliton if and only if GL ( n, R ) · µ is a distinguished orbit. Moreover, there is a curve of metrics between the distinguished metric and thesolsoliton metric which preserves their algebraic isometry groups.In particular, when S µ is unimodular, we see that these two Riemannian Lie groups have the sameisometry groups. The claims in the first paragraph follow from the above observations. The last claim will be proved inTheorem 5.8. 11 emark 4.7. There are solvable groups which admit a distinguished metric, but cannot admit a solsoliton.For example, if n is a non-abelian Einstein nilradical and a ⊂ Der ( n ) is an abelian subalgebra of skew-symmetric endomorphisms, then S µ with s µ = a ⋉ n cannot admit a solsoliton but does admit a distinguishedmetric. See Theorem 5.11 and [Lau03, Theorem 4.7]. The following has been shown for nilpotent groups in [Nik08a] and [Jab08a], but has not appeared in theliterature for solvable groups in general. The corollary which follows is of particular interest and it would beinteresting to know if there is an analogous statement for compact homogeneous spaces admitting Einsteinmetrics. Theorem 4.8. Let S µ be a solvable Lie group whose Lie algebra s µ = s µ + s µ is a direct sum of ideals.Then S µ admits a non-flat solsoliton, resp. flat, metric if and only if both S µ and S µ admit non-flatsolsoliton, resp. flat, metrics. Corollary 4.9. Let S µ be a solvable Lie group whose Lie algebra s µ = s µ + s µ is a direct sum of ideals.Then S µ admits an Einstein metric if and only if both S µ and S µ admit Einstein metrics of the same sign.Proof of theorem. We prove the case that the solsoliton is not flat. The flat case is similar and we leave itto the reader.One direction is trivial. Recall, a non-flat solsoliton with Ric = cId + D satisfies c < S µ i admit solsolitons satisfying Ric µ i = c i Id + D i , then one just needs to rescale so that c = c . Endow s µ with the product metric, i.e. the s µ i are orthogonal and the restriction to s µ i is the aforementioned metric.Then s µ , with µ = µ + µ , is a solsoliton satisfying Ric µ = Ric µ ⊕ Ric µ = c Id + ( D ⊕ D ).We now show the converse. Recall that S µ admitting a solsoliton implies the orbit GL n R · µ is distinguishedby Proposition 4.4. However, this implies the orbits GL n i R · µ i ⊂ ∧ ( R n i ) ⊗ R n i are distinguished, where n i = dim s µ i . (This has been proven in [Jab08a, Theorem 4.5] for nilpotent groups. However, the proofthere only uses the fact that the orbits are distinguished and works in this setting with no modifications.)Assume now that µ i are the distinguished points and write s µ i = a i ⊕ n i where n i is the nilradical and a i is a reductive subalgebra (cf. [Lau03, Theorem 4.7]). As s µ i are distinguished, the nilradicals n i admitnilsolitons by [Lau03, Theorem 4.7].Write s µ = s µ + s µ = ( a + a ) + ( n + n ). As s µ is solvable, we see that the reductive subalgebra a = a + a is abelian and hence each a i is abelian. Furthermore, for any A ∈ a , we see that ad A : n → n has no purely imaginary eigenvalues by the observations in the proof of Proposition 4.4. Thus, the solvablegroups S µ i admit solsoliton metrics by either the observations in the proof of Proposition 4.4 or Theorem5.11. Remark. We point out for the concerned reader that the proof of Theorem 5.11 does not depend on theprevious theorem.Proof of corollary. The proof of the corollary follows immediately from the proof of the theorem and thefact that isomorphic distinguished points must be isometric. More precisely, isomorphic distinguished pointslie in the same O ( n )-orbit, see Theorem 5.3. In this section we analyze the negative gradient flow of F = || m || as the critical points of this function havegeometric significance, see Theorems 3.3 and 4.6.Let G be an θ -stable subgroup of GL ( n, R ), i.e. G is stable under the transpose operation. Let K G denote the set of fixed points of θ ( g ) = ( g t ) − (cf. Section 3). Denote the moment map of this group actionby m G and consider the function F = || m G || with critical set C G . Denote the negative gradient flow of F by ϕ t ; in the notation of Definition 4.1 ϕ t ( µ ) = µ t . In the following way we consider limits of this flow.12 efinition 5.1. The ω -limit set of ϕ t ( p ) ⊆ V is the set { q ∈ V | ϕ t n ( p ) → q for some sequence t n →∞ in R } . We denote this set by ω ( p ) . Proposition 5.2. [Sja98] The omega limit set ω ( p ) is a single point. The uniqueness of limits is a strong result and is due to the fact that F = || m || is real analytic, K -invariant, and that C G ∩ { sphere of radius || p ||} ∩ G · p ⊂ K · p (see theorem below). As the limit iswell-defined, we will denote it by ω ( p ) = ϕ ∞ ( p ). We point out that many of the following results can beproven without knowing that there is a unique point in the limit set. Theorem 5.3. [Jab08a] Consider p ∈ C G . Theni. F ( p ) is a minimum of F restricted to G · p ,ii. C G ∩ { sphere of radius || p ||} ∩ G · p ⊂ K · p , andiii. ω ( G · p ) ⊂ K G · p , i.e. ϕ ∞ ( gp ) ∈ K G · p for all g ∈ G . The first two statements originally appeared in [KN78] for complex representations and in [Mar01] forreal representations. In this way, we see that orbits containing critical points of F = || m G || are stable inthe sense that the critical set is a global attractor of the negative gradient flow along the entire orbit.In the setting of GL n R acting on V = ∧ ( R n ) ∗ ⊗ R n , if µ ∈ V is the Lie bracket of a solvable Lie groupadmitting a solsoliton, then the orbit GL n R · µ contains a critical point of the function F = || m || (seeProposition 4.4). We use this below to recover Einstein and solsoliton metrics. Finding Einstein Metrics Using the above observations, we now have a procedure for recovering an Einstein, or solsoliton, metric ona solvable Lie group when it exists. Proposition 5.4. Let G be a solvable Lie group which admits a non-flat Einstein or solsoliton metric. Thesolsoliton metric may be obtained by following two consecutive curves of metrics.Let h· , ·i be any initial metric. The first curve h· , ·i t , t ∈ [0 , , goes from the initial metric to a so-called‘distinguished metric’ via a negative gradient flow. The second curve h· , ·i t , t ∈ [1 , , joins the distinguishedmetric to the solsoliton metric by simply modifying the metric on the orthogonal complement a of the nilradical n . Remark. A similar result holds for solvable Lie groups admitting flat metrics. Here one just uses the firstcurve described above, cf. Proposition 4.4.Proof. We realize this theorem by evolving the bracket instead of the metric. Identify the metric Lie algebra { g , h · i} with s µ for some µ ∈ V . The first curve comes from flowing µ along the negative gradient flowof F = || m || . This converges within the isomorphism class GL n R · µ as the orbit is distinguished (seeProposition 4.4). This limit is a distinguished point.The second curve is realized by changing the metric on a × a as in the proof of Proposition 4.4: (cid:18) − c (cid:19) (cid:20) − t tr ad A ◦ ad A t + ( t − tr S ( ad A ) (cid:21) for t ∈ [1 , Remark 5.5. In the event that the solsoliton is a flat Einstein metric, the second curve simply rescales theinitial metric, however, when the solsoliton is not a flat Einstein metric, then the second curve consists ofgenuinely distinct metrics (i.e., non-homothetic metrics). olitons and isometry groups The following is an immediate consequence of Theorem 5.3. Corollary 5.6. Consider G a real algebraic reductive θ -stable subgroup of GL ( n, R ) acting on V . Let G · p be a distinguished orbit and ϕ ∞ ( p ) as above. Then K p ⊂ K ϕ ∞ ( p ) , where K = G θ and K q is the stabilizersubgroup at q .Proof. This follows from the K -equivariance of m G and the uniqueness of integral curves of the negativegradient flow of || m G || . In fact, one has K p ⊂ K ϕ t ( p ) and the result follows by taking the limit. Theorem 5.7. [Jab09] Consider a θ -stable subgroup G of GL ( n, R ) acting on V (as in Section 3). Suppose H is a θ -stable group of automorphisms of µ ∈ V . Consider the centralizer of H in GZ G ( H ) = { g ∈ G | gh = hg for all h ∈ H, g ∈ G } Then Z G ( H ) is reductive, θ -stable and G · µ is a distinguished orbit if and only if Z G ( H ) · µ is a distinguishedorbit. Moreover, along the orbit Z G ( H ) · µ , m G = m Z G ( H ) . Remark. In the above, there is no ambiguity as to the meaning of distinguished since m G = m Z G ( H ) alongthe subset Z G ( H ) · µ .The group H being a group of automorphisms means precisely that H is a subgroup of the stabilizer of GL ( n, R ) at µ , and H being θ -stable automatically makes H reductive. This theorem has been used to construct continuous families of Einstein nilradicals and non-Einsteinnilradicals (cf. [Jab09]). We use this theorem to narrow our search for soliton metrics and to help prove thefollowing. Theorem 5.8. Let S be a completely solvable unimodular group admitting a solsoliton metric. Let g be anyleft-invariant metric. Then there exists a left-invariant soliton metric g ′ such that Isom ( S, g ) ⊂ Isom ( S, g ′ ) ,as groups. Corollary 5.9. Let N be an Einstein nilradical. Let g be any left-invariant metric. Then there exists aleft-invariant soliton metric g ′ such that Isom ( N, g ) ⊂ Isom ( N, g ′ ) , as groups. Remark 5.10. In this way, we see that these soliton metrics are the most symmetric (left-invariant) metricthat such nilpotent and solvable groups can admit.Proof. Recall that a completely solvable unimodular Lie group S µ admits a solsoliton metric if and onlyif GL ( n, R ) · µ is a distinguished orbit (Theorem 4.6). To show that such metrics have maximal isometrygroups, we use an intermediate metric, a distinguished metric (i.e. critical point of F = || m || ), in which theisometry group embeds and then show that this metric has the same isometry group as a particular choiceof soliton metric (cf. Section 4).Recall that the isometry group of a completely solvable unimodular group is its algebraic isometry group,i.e. Isom ( S µ ) = S µ ⋊ ( Aut ( µ ) ∩ O ( h , i ). Given g ∈ GL ( n, R ), Aut ( g ∗ µ ) = gAut ( µ ) g − and the orthogonalgroup O ( ( g − ) ∗ h , i ) = g − O ( h , i ) g , as ( g − ) ∗ h· , ·i = h g · , g ·i .The following metric Lie algebras are isometric { R n , g ∗ µ, h· , ·i} ≃ { R n , µ, ( g − ) ∗ h· , ·i} see Section 2, and the corresponding Riemmanian solvable Lie groups are isometric { S g ∗ µ , h· , ·i} ≃ { S µ , ( g − ) ∗ h· , ·i} At e ∈ S µ , the isometry group of { S µ , ( g − ) ∗ h· , ·i} has isotropy subgroup Aut ( µ ) ∩ O ( ( g − ) ∗ h , i ) = g − ( Aut ( g ∗ µ ) ∩ O ( h , i ) ) g tep 1. Let S µ be a Riemannian solvable group which admits a solsoliton metric. Let H = Aut ( µ ) ∩ O ( h , i ); this subgroup is θ -stable. By Theorem 4.6 the orbit GL ( n, R ) · µ is distinguished and by Theorem5.7 the orbit Z G ( H ) · µ actually contains the limit µ ∞ of the negative gradient flow of F = || m || . Let g ∈ Z g ( H ) be such that g · µ = µ ∞ .By Corollary 5.6, we see that Aut ( µ ) ∩ O ( h , i ) = K µ ⊂ K g · µ = Aut ( g ∗ µ ) ∩ O ( h , i )where K = O ( n, R ). Using the fact that g ∈ Z G ( H ), we obtain Aut ( µ ) ∩ O ( h , i ) = g − ( Aut ( µ ) ∩ O ( h , i ) ) g ⊂ g − ( Aut ( g ∗ µ ) ∩ O ( h , i ) ) g = Aut ( µ ) ∩ O ( ( g − ) ∗ h , i )As the underlying Lie group structure of { S µ , h , i} and { S µ , ( g − ) ∗ h , i} is the same, have have Isom ( S µ , h , i ) ⊂ Isom ( S µ , ( g − ) ∗ h , i )as Lie groups. Step 2. So far we have imbedded the isometry group of S µ into the isometry group of a distinguishedmetric S µ ′ (these are isomorphic as Lie groups). Write s µ ′ = a ⋉ n . We have already observed that themetric on s µ ′ can be transformed into a solsoliton metric by simply rescaling the metric on a and this doesnot change the isometry groups, see Remark 4.5. This completes the proof. Characterization of solvable algebras admitting Einstein and solsoliton metrics Theorem 5.11. Let n be an Einstein nilradical and denote the algebra of derivations by Der ( n ) . Let a ⊂ Der ( n ) be an abelian reductive subgroup. If no element of a has only purely imaginary eigenvalues, then s = a ⋉ n admits a solsoliton metric. Moreover, every solvable algebra admitting a non-flat solsoliton metricarises this way. Corollary 5.12. If in addition to the above hypotheses, a contains some pre-Einstein derivation D , then s = a ⋉ n admits a negative Einstein metric. Moreover, every solvable algebra admitting a negative Einsteinmetric arises this way. The above characterization of solvable Lie groups admitting negative Einstein metrics is essentially acombination of the above characterization of solsolitons together with Lauret’s structural results, cf. Theorem2.8. As such, we leave the proof of the corollary to the reader. The definition of pre-Einstein derivation maybe found in Definition 8.1.Below we will prove that the algebras described above admit solsoliton metrics. The fact that all solsoli-tons have such a rigid algebraic structure is the work of Lauret, see Theorem 2.8. Proof of Theorem 5.11. Take a as above and consider it as a subalgebra of gl ( n µ ). Let A be the connectedsubgroup of GL ( n µ ) with Lie algebra a . Denote by A the Zariski closure of A in GL ( n µ ) (i.e., the smallestalgebraic group containing A ) and its Lie algebra by a . As Aut ( n µ ) is an algebraic group, A ⊂ Aut ( n µ ).Moreover, A is abelian and reductive. The fact that A is abelian follows immediately from being the closureof an abelian group. To see that this group is reductive, one can ‘diagonalize’ a to see that A is a subgroupof a torus (abelian, reductive) of GL ( n µ ) and hence has no non-trivial nilpotent elements.It is a classical fact that there exists g ∈ GL ( n, R ) such that g a g − is θ -stable since A is both algebraicand reductive, see [Mos55]. Now a = g a g − is a reductive, abelian subalgebra of Der ( g ∗ µ ) and a ⋉ n µ ≃ a ⋉ n g ∗ µ via the isomorphism which is the identity on a and g on n µ . The nilpotent Lie group N g ∗ µ is an Einsteinnilradical if and only if N µ is so, as they are isomorphic.15e will apply Theorem 5.7 to the subgroup A = gAg − ⊂ GL ( n, R ) with Lie algebra a = gag − . Thisgroup is θ -stable as its Lie algebra is so. Let µ = g ∗ µ and consider the limit µ ∞ of the flow µ t . TheRiemannian nilpotent Lie group N µ ∞ is a nilsoliton and µ ∞ = g ′∗ µ for some g ′ ∈ Z GL ( n, R ) ( A ). As g ′ commutes with a we see that the following solvable algebras are isomorphic a ⋉ N µ ≃ a ⋉ N µ ≃ g ′ a g ′− ⋉ N g ′∗ µ = a ⋉ N g ′∗ µ = a ⋉ N µ ∞ but the last metric algebra satisfies all the criteria of Theorem 2.8 to be a solsoliton metric Lie algebra. Construction of the finer subgroup I G ( H ) ⊂ Z G ( H ) ⊂ G In the above proofs, one can use a smaller subgroup of Z G ( H ) whose orbit will contain critical points of F = || m G || . This group will be used in Section 9 to construct the algorithm that determines when anilpotent Lie group is an Einstein nilradical. Proposition 5.13. Let H be a θ -stable subgroup of Aut ( µ ) , as in Theorem 5.7. Assume G ∩ Aut ( µ ) = H ,i.e. the stabilizer at µ of the group G acting on V = ∧ ( R n ) ∗ ⊗ R n is H .There exists a real algebraic reductive subgroup I G ( H ) of G such that Z G ( H ) = I G ( H )( Z G ( H ) ∩ H ) where Z G ( H ) ∩ H is the stabilizer subgroup of Z G ( H ) at µ and the Lie algebra of I G ( H ) satisfies i G ( H ) = { X ∈ z G ( H ) | tr ( XY ) = 0 for all Y ∈ z G ( H ) ∩ h } Moreover, the orbits coincide, i.e. I G ( H ) · µ = Z G ( H ) · µ . Here the Lie algebra z G ( H ) = z g ( h ) of Z G ( H ) is the commutator of h in g . One can see by directcalculation that i G ( H ) is a Lie subalgebra. We show that its corresponding Lie subgroup of GL ( n, R ) is analgebraic group so that we can exploit the methods of Section 3. Definition 5.14. An element X ∈ gl ( n, R ) will be called algebraic if it is tangent to a real algebraic 1-parameter subgroup of GL ( n, R ) . More generally, a Lie subalgebra will be called algebraic if it is tangent toan algebraic subgroup of GL ( n, R ) . An element X ∈ g is called reductive if it is semisimple (over C ). We observe that if G ⊂ GL ( n, R ) is anyreal reductive algebraic subgroup, the set of reductive algebraic elements of g is dense. As we are considering G which are θ -stable, the following bilinear form is an inner product on g h X, Y i = tr ( XY t )Given a θ -stable element α ∈ g (i.e. α is symmetric or skew-symmetric), we define the subalgebra g α ⊖ α = { X ∈ g α | tr ( Xα t ) = 0 } where g α = { X ∈ g | [ X, α ] = 0 } . Since α t = ± α , it follows that g α ⊖ α is θ -stable and an ideal of g α . Lemma 5.15. The subalgebra g α ⊖ α is an algebraic Lie subalgebra. From this lemma, the proposition above quickly follows. To see this, observe that z g ( h ) ∩ h is θ -stableand decompose z g ( h ) ∩ h = ( z ∩ h ) k ⊕ ( z ∩ h ) p into its Cartan decomposition. All the elements contained in( z ∩ h ) k and ( z ∩ h ) p are algebraic reductive elements. Now apply the above lemma to all these algebraicreductive elements and use the fact that the intersection of algebraic groups is algebraic. Proof of lemma. The cases of α symmetric and skew-symmetric must be handled separately. Case: α symmetric. Every such α is conjugate via O ( n, R ) to a diagonal matrix. As the above innerproduct is Ad O ( n, R ) invariant and the conjugate of an algebraic group is algebraic, we may reduce to the16ase that α is diagonal. Also, we may reduce to the case G = GL ( n, R ) as the intersection of algebraicgroups is algebraic.Further more, we may assume (via conjugation by O ( n, R )) that the eigenvalues are weakly increasing: α = diag { a , . . . , a , . . . , a k , . . . , a k } . The eigenvalues a i are rational as α is algebraic. Now the subalgebra g α consists of block diagonal matrices gl ( n , R ) × · · · × gl ( n k , R ). This is clearly an algebraic Lie algebrawhose Lie group G α consists of the block matrices which are invertible.The condition X ∈ g α is now Σ a i X i = 0 where X = blockdiag { X , . . . , X k } . Write g ∈ G α as a blockdiagonal matrix g = blockdiag { g , . . . , g k } . Then the algebraic group with Lie algebra g α ⊖ α is { g ∈ G α | Π det ( g i ) qa i = 1 } where q is the common integer such that qa i ∈ Z for all i = 1 , . . . , k . Case: α skew-symmetric. To prove the result in this case, we reduce to the above case and use complexalgebraic groups. We will construct a complex algebraic group whose intersection with GL ( n, R ) is thedesired Lie group. This Lie group will be algebraic as it is the intersection of algebraic groups. We refer thereader to [Whi57] for an introduction to the relationship between real and complex varieties.Observe that the above work for α symmetric could have been carried out over C . Consider g C α = { X ∈ gl ( n, C ) | [ X, α ] = 0 } . Observe that iα has real eigenvalues (which may be assumed to be rational as above)and that g C α = g C iα and g C α ⊖ α = g C iα ⊖ iα . By conjugating with U ( n ) ⊂ GL ( n, C ), we may assume iα isdiagonal. Following the above work, but with complex groups instead of real, we have a complex algebraicgroup over g C α ⊖ α = g C iα ⊖ iα . Moreover, this group intersected with GL ( n, R ) is a real Lie group with thedesired Lie algebra. Counting dimensions, we see that the real points of this complex algebraic group areZariksi dense and hence this Lie group is real algebraic. In this section we apply the above results to compact quotients of nilpotent Lie groups that admit solitonmetrics. Definition 6.1. Let ( M, g ) = (Γ \ N, g ) be a compact nilmanifold where Γ ⊂ N ⊂ Isom ( N, g ) , g is a left-invariant metric, and the metric on M is the induced metric coming from N . The metric g on Γ \ N is calleda local nilsoliton if ( N, g ) is a nilsoliton. As in the case of Ricci solitons on nilpotent Lie groups, local nilsolitons may be characterized as criticalpoints of a functional restricted to the set of locally N -invariant metrics. In fact, these metrics are minimaof the function F ( g ) = R M tr Ric g dV ol g R M sc ( g ) dV ol g , restricted to the set of locally N -invariant metrics, see [Jab10] forthis point and more analysis on this functional. Remark 6.2. While nilsolitons are unique on a simply connected nilpotent Lie group (up to rescaling andisometry), this does not remain true for local nilsolitons on compact quotients. On compact nilmanifolds, local nilsoliton metrics are the most symmetric among all locally-left-invariantmetrics (cf. Corollary 5.9). Theorem 6.3. Consider M = Γ \ N endowed with a locally left-invariant metric g where N admits a nil-soliton. Then there exists a local soliton g ′ on M such that Isom ( M, g ) ⊂ Isom ( M, g ′ ) .Proof. The proof reduces to the corresponding statement on simply-connected covers: Corollary 5.9.Let φ ∈ Isom ( M, g ) and consider the Riemannian quotient π : N → Γ \ N . The map φ lifts to adiffeomorphism φ : N → N such that π ◦ φ = φ ◦ π . As (Γ \ N, g ) and ( N, g ) are locally isometric via π and φ is an isometry, we have φ ∈ Isom ( N, g ). Conversely, every isometry of M arises from φ ∈ Isom ( N, g )satisfying the condition φ (Γ n ) = Γ φ ( n ) for all n ∈ N (6.1)17bserve that this condition is independent of any metric data.By Corollary 5.9, there exists a nilsoliton g ′ on N such that φ is an isometry of ( N, g ′ ) and this choiceof g ′ holds for all φ . As the above relation (6.1) still holds, the diffeomorphism φ : M → M is an isometryrelative to g ′ . Theorem 6.4. The existence of a local soliton depends only on the fundamental group.Proof. This is a consequence of the classical fact that the fundamental group Γ completely determines thenilpotent group N . More precisely, let Γ , Γ be the fundamental groups of compact nilmanifolds Γ \ N andΓ \ N . If φ : Γ → Γ is an isomorphism of abstract groups, there exists an isomorphism Φ : N → N ofLie groups such that φ = Φ | Γ , see [Rag72, Theorem 2.11 and Corollary 2].The claim now follows from the algorithm of Section 9 which shows that the existence of nilsolitons on N i is a property of the underlying Lie algebra. Remark. The above theorems on compact nilmanifolds hold for infranilmanifolds as every infranilmanifoldis finitely covered by a compact nilmanifold. V To refine our analysis of the Riemannian Lie groups S µ , and the function F = || m || , we stratify the space V . Using this stratification, we obtain a decomposition of the automorphism group Aut ( µ ) which aids inthe construction of algorithms to determine the existence of soliton metrics, see Lemma 7.7.Denote the critical set of F = || m || by C . Theorem 7.1 ([Lau07, LW07]) . There exists a finite subset B ⊂ g , and for each β ∈ B a GL n R -invariantsmooth submanifold S β ⊂ V (a stratum), such that V \{ } = G β ∈ B S β This stratification satisfies S β − S β = F || β ′ || > || β || S β ′ . Additionally, C = F β ∈ B C β where C β ⊂ S β are thecritical points with critical value M β = || β || .For µ ∈ S β , following conditions are satisfied:i. h [ β, D ] , D i ≥ for all D ∈ Der ( µ ) with equality if and only if [ β, D ] = 0 .ii. β + || β || I is positive definite for all β ∈ B such that S β ∩ N 6 = ∅ .iii. || β || ≤ || m ( µ ) || with equality if and only if m ( µ ) is conjugate to β under O ( n ) .iv. tr βD = 0 for all D ∈ Der ( µ ) .v. h π ( β + || β || I ) µ, µ i ≥ with equality if and only if β + || β || I ∈ Der ( µ ) . Remark. The finite subset B ⊂ g consists of diagonal elements, with positive entries on the diagonal whichare (weakly) increasing. We will not reconstruct this stratification and direct the interested reader to those works above. Instead,we describe the necessary properties below that suit our needs. This stratification is the real analog ofwell-known stratifications in Geometric Invariant Theory over algebraically closed fields. For representationsof complex reductive groups, such stratifications coincide with a Morse theoretic stratification coming from F = || m || . This result remains true in the setting of real representations and is an immediately consequenceof Theorem 7.4. 18iven α ∈ diag ⊂ gl ( n, R ), the diagonal matrices, we define the following groups G α = { g ∈ GL ( n, R ) | gαg − = α } U α = { g ∈ GL ( n, R ) | exp ( tα ) g exp ( − tα ) → e as t → −∞} P α = G α U α P α is the parabolic associated to α with unipotent radical U α and reductive complement G α . As α t = α , G α is θ -stable and has a Cartan decomposition G α = K α exp ( p α ) (cf. definition of Cartan involution in Section3). When α ∈ B , the eigenvales of α are weakly increasing and the group G α consists of block diagonalmatrices (which commute with α ) while U α is the group of lower triangular elements beneath the blocks of G α .Define the subgroup H α as the group with Lie algebra h α = { X ∈ g α | tr ( Xα ) = 0 } ; this is actually analgebraic group. In the following proposition, we maintain the notation from [Lau07]. Definition 7.2. A point v ∈ V is called H α -stable if H α · v . Proposition 7.3 (Lauret) . Given β ∈ B , there exist subsets Z β and Y β with the following properties:i. Y β is P β -invariant, Y ssβ = Y β ∩ S β consists of H β -semi-stable points and S β = O ( n ) Y ssβ ii. For y ∈ Y β , { g ∈ GL ( n, R ) | g · y ∈ Y β } = P β iii. Z β = { v ∈ Y β | π ( β ) v = || β || v } , Z β is G β -invariant, S β ∩ Z β = Z ssβ (the H β -semi-stable points of Z β ) and S β = GL ( n, R ) Z ssβ = O ( n ) P β Z ssβ iv. The H β orbits intersecting Z β ∩ C β are all closed. Remark. Part ii above does not appear in [Lau07]. However, one can show this immediately just as in thecomplex case (cf. Lemma 13.4 of [Kir84]). The following theorem and its proof have appeared in a more general form in [HSS08]. In our setting, ashort proof is readily obtained, and so we include one for completeness. Theorem 7.4 (Heinzner-Schwarz-St¨otzel) . Consider µ ∈ S β . There exists a unique GL ( n, R ) -orbit in GL ( n, R ) · µ ∩ S β intersecting C β . The closed orbits in S β are precisely those intersecting C β .Proof. Take µ ∈ S β . As S β = O ( n ) Y ssβ , we may assume µ ∈ Y β and thus GL ( n, R ) · µ = O ( n ) P β µ . Since O ( n ) is compact, we see that GL ( n, R ) · µ = O ( n ) P β · µ .Consider any point y ∈ Y β and the curve exp ( tβ ) · y with limit y −∞ as t → −∞ (this limit exists by thedefinition of Y β ). Observe that exp ( tβ )( P β · y ) → G β · y −∞ as t → −∞ . To see this, write g ∈ P β as g = g g with g ∈ G β and g ∈ U β , then exp ( tβ ) · gy = g exp ( tβ ) g exp ( − tβ ) exp ( tβ ) · y → g e y −∞ .Now take y ∈ GL ( n, R ) · µ ∩ C β . By the above theorem, there exists k ∈ O ( n ) such that k · y ∈ Z β . Sowe may assume k = e and y ∈ GL ( n, R ) · µ ∩ C β ∩ Z β . This point is fixed by exp ( tβ ) and we see that G β · y ⊂ G β · µ −∞ by applying exp ( tβ ) and letting t → −∞ .As y and µ −∞ are both eigenvectors for β , we see that under the map V → P V , v [ v ], H β · [ y ] ⊂ H β [ µ −∞ ]. Now, as H β · y is closed, the uniqueness result follows from [RS90].The above theorem answers Question 3.4. Corollary 7.5. Let N µ be an Einstein nilradical. Let µ ∞ denote the limit point of the negative gradientflow of the function F = || m || starting at µ . Then µ ∞ is contained in the orbit GL n R · µ ; that is, N µ and N µ ∞ are isomorphic Lie groups.Proof. This follows immediately from the fact that the limit µ ∞ ∈ S β ∩ GL ( n, R ) · µ and the orbit GL ( n, R ) · µ is closed in S β . 19 utomorphisms of Einstein Nilradicals Given that nilsolitons are precisely the critical points of F = || m || (Theorem 3.3) we have the followingdecomposition of Aut ( µ ). The following decomposition holds more generally with µ being the critical pointof || m || and Aut ( µ ) being replaced by the stabilizer subgroup of an action. In particular, there is a similardecomposition of the automorphism group of a solvable Lie group admitting a solsoliton. Proposition 7.6. Let µ ∈ N be a soliton in the stratum S β . Let G β be the centralizer of β in GL n R and U β = { g ∈ GL n R | exp ( tβ ) gexp ( − tβ ) → e as t → −∞} . Then the automorphism group of N µ decomposesas Aut ( µ ) = G β U β = K β exp ( p β ) U β where G β = G β ∩ Aut , K β = O ( n ) ∩ G β ∩ Aut , exp ( p β ) = exp ( symm ( n )) ∩ G β ∩ Aut , U β = U β ∩ Aut .Proof. This result follows quickly from Proposition 7.3. Let µ ∈ S β be the nilsoliton of interest, where S β isthe stratum defined above. By considering O ( n ) translates of µ , we may assume that µ ∈ Z β .Let g ∈ Aut ( µ ). Since g · µ = µ ∈ Z β ⊂ Y β , g ∈ P β by Part ii. of Proposition 7.3, and we may write g = g g where g ∈ G β and g ∈ U β . Observe that exp ( tβ ) g exp ( − tβ ) also stabilizes µ and letting t → −∞ we see that g ∈ Aut ( µ ) and hence g ∈ Aut ( µ ). This shows Aut ( µ ) = G β U β .Given g ∈ G β , write g = k exp ( X ) where k ∈ K β and X ∈ p β ; this is possible as G β is stable underthe transpose. Observe, || m ( µ ) || = || m ( g · µ ) || = || m ( exp ( X ) · µ ) || and by [NM84, Lemma 7.2] we see that exp ( X ) · µ = µ . Thus exp ( X ) , k ∈ Aut ( µ ) and the theorem is proved.There is an analogous decomposition of Der ( µ ) as above. The following is presented for later use. Lemma 7.7. Take a nilsoliton N µ with Einstein derivation β and derivation algebra Der ( µ ) = k β ⊕ p β ⊕ u β .Every element of the form β + X, with X ∈ u β is semi-simple (i.e. diagonalizable).Sketch of proof. The proof of this fact is analogous to showing that any upper triangular matrix with non-zero distinct entries on the diagonal can be diagonalized. One carries out similar computations in this case(as the entries of β are all positive and u β has an appropriate block structure) to show that β + X can beconjugated to β via U β . Let n µ be an Einstein nilradical with Ricci tensor Ric µ = cId + D , for some c ∈ R and D ∈ Der ( µ ). Thisderivation is semisimple with real, positive eigenvalues and satisfies the condition tr ( Dψ ) = − c tr ψ for any ψ ∈ Der ( µ ). The derivation D satisfying Ric = cId + D is called an Einstein derivation as itsexistence is necessary for solvable extensions of n µ to admit an Einstein metric. As such, we make thefollowing definition. Definition 8.1. A derivation φ of a Lie algebra s µ is called pre-Einstein, if it is semisimple, with alleigenvalues real, and tr ( φψ ) = tr ψ , for any ψ ∈ Der ( µ ) (8.1)The so-called Einstein derivation D gives the pre-Einstein derivation φ = D − c , and viceversa. Remarkably,determining the pre-Einstein derivation almost completely determines when a nilpotent Lie algebra admitsa nilsoliton metric. This derivation first appeared in [Nik08a]. Theorem 8.2. [Nik08a] . (a) Any Lie algebra s µ admits a pre-Einstein derivation φ µ .(b) The derivation φ µ is determined uniquely up to automorphism of s µ .(c) All the eigenvalues of φ µ are rational numbers.ii. Let n µ be a nilpotent Lie algebra, with φ a pre-Einstein derivation. If n µ is an Einstein nilradical, thenits Einstein derivation is positively proportional to φ . Remark. Part ii. above is particularly useful as it can be difficult to determine, a priori, which stratum µ belongs to (cf. Section 7). Moreover, we will see that determining the pre-Einstein derivation reduces tosolving a system of linear equations (the condition of semi-simplicity can be discarded, cf. Proposition 8.4). Let n µ be a nilpotent Lie algebra with a choice of pre-Einstein derivation φ µ . Associated to φ µ we havethe following subalgebra g µ = z ( φ µ ) ∩ ker ( T ) ⊂ sl ( n, R )where z ( φ µ ) is the centralizer of φ µ and T ( A ) = tr ( Aφ µ ). Let G φ µ ⊂ SL ( n, R ) be the Lie group with algebra g φ µ , this is an algebraic group. Theorem 8.3. [Nik08a] For a nilpotent Lie algebra n µ with a pre-Einstein derivation φ , the followingconditions are equivalent:i. n µ is an Einstein nilradicalii. the orbit G φ · µ ⊂ V is closed In this way, we see that the property of a nilpotent Lie group being an Einstein nilradical is intrinsic toits Lie algebra. We will build on this result to obtain an algorithm which determines the condition of beingan Einstein nilradical using only local data, see Section 9. To simplify our work, we present the followingreduction. Proposition 8.4. If n µ is an Einstein nilradical, then any solution to Equation (8.1) will automatically bea pre-Einstein derivation, i.e. it is automatically semi-simple with real, positive eigenvalues. Remark 8.5. In this way, we see that if a solution to Equation (8.1) is not semi-simple, then the nilpotentgroup in question is not an Einstein nilradical.Proof. The proof amounts to analyzing Nikolayevki’s proofs and combining those details with Lemma 7.7.For the sake of completeness, we present Nikolayevski’s proof of existence and uniqueness (up to conjugationin Aut ) of the pre-Einstein derivation.First we find one pre-Einstein derivation. Let s µ be a Lie algebra and denote by Der ( µ ) its algebra ofderivations; this is an algebraic Lie algebra (meaning it is the Lie algebra of an algebraic group). Considera Levi decomposition Der ( µ ) = s ⊕ t ⊕ n where t ⊕ n is the radical of Der ( µ ), s is semisimple, and n is theset of nilpotent elements (the nilradical) of t ⊕ n , t is a torus (abelian subalgebra with semisimple elements),and [ t , s ] = 0.Recall, for ψ ∈ gl ( n, R ) a semisimple endomorphism, there exist semisimple endomorphisms ψ R and ψ i R (the real and imaginary parts) which have real, resp. purely imaginary, eigenvalues such that ψ = ψ R + ψ i R and all three endomorphisms commute. Moreover, the subspaces t c = { ψ i R | ψ ∈ t } and t s = { ψ R | φ ∈ t } are the compact and the fully R-reducible tori (the elements of t s are simultaneously diagonalizable) with t s ⊕ t c = t .We will find a pre-Einstein derivation contained in t s . Consider the quadratic form b on Der ( µ ) definedby b ( ψ , ψ ) = tr ( ψ ψ ). It is a general fact that n is in the kernel of this quadratic form, hence b ( t , ψ ) = 0 = tr ( ψ )for any ψ ∈ n . Using the ad-invariance of b (that is, b ( ψ , [ ψ , ψ ]) = b ([ ψ , ψ ] , ψ )) and that s = [ s , s ] issemisimple, we see that b ( t , ψ ) = 0 = tr ( ψ )21or any ψ ∈ s . Thus it suffices to solve Equation (8.1) with φ, ψ ∈ t . Additionally, observe that b ( t s , ψ ) = 0 = tr ( ψ )for any ψ ∈ t c . Lastly, as the quadratic form b restricted to t s is positive definite, the existence (anduniqueness in t ) follows.To obtain the uniqueness of the pre-Einstein derivation up to conjugation in Aut , Nikolayevski exploitsa theorem of Mostow [Mos56, Theorem 4.1] which says that all fully reducible subalgebras of Der ( µ ) areconjugate via an inner automorphism of Der ( µ ). Lastly, as the center of a reducible algebra is uniquelydefined, we have the desired result.Now we analyze this proof to study all solutions to Equation (8.1). Let A ∈ s ⊕ t be a solution to tr ( Aψ ) = 0 for all ψ ∈ Der ( µ ). We will show that A = 0. To see this, first assume that our Lie algebra n µ is endowed with an inner product so that s ⊕ t is stable under the transpose operation. This is alwayspossible; when n µ is an Einstein nilradical such a metric is explicitly given in Proposition 7.6. Using thisinner product, ψ = A t ∈ Der ( µ ) and 0 = tr ( Aψ ) = tr ( AA t ) implies A = 0.Let φ ∈ t be a pre-Einstein derivation of n µ , the above work shows that any solution to Equation (8.1)is of the form φ + X where X ∈ n (the nilpotent part of the radical of Der ( µ )). And applying Lemma 7.7we are finished. In this section, we demonstrate how the existence of a nilsoliton on a nilpotent Lie group can be read offfrom local data. More precisely, let N be a nilpotent Lie group of interest with Lie algebra n . To determineif N admits a nilsoliton, one only needs to analyze Der ( n ) and certain infinitesimal deformations of anyinitial left-invariant metric on N . Theorem 9.1. The existence of a nilsoliton metric on a nilpotent Lie group N is intrinsic to the underlyingLie algebra n . More precisely, one can determine the existence of such a metric by analyzing the derivationalgebra Der ( n ) and infinitesimal deformations of any initial metric on n . Remark 9.2. The existence of a nilsoliton being intrinsic to the Lie algebra was first shown by Nikolayevsky[Nik08a]. Here it was shown that the existence of such a metric is equivalent to an orbit of a particularreductive group being closed in the space of Lie brackets (see Theorem 8.3). However, it was not shown thatthis could be determined by measuring local data.Before Nikolayevsky’s result, it was shown by Lauret [Lau01a] that the existence of such a metric isequivalent to the full GL n R -orbit in the space of Lie brackets being so-called distinguished . However, it wasnot known before the present work that this condition may be determined locally. Algorithm to determine if N is an Einstein nilradical Step 1: Find a solution φ to tr ( φψ ) = tr ( ψ ) for all ψ ∈ Der ( n )If the solution is φ is not semisimple (i.e. diagonalizable) with (positive) real eigenvalues then stop, n isnot an Einstein nilradical.If φ is semisimple with (positive) real eigenvalues, then continue; this is a pre-Einstein derivation of n (cf.Definition 8.1 and Proposition 8.4). (Remark: positivity of the eigenvalues will be automatic if the followingsteps are valid.) Step 2: Consider the subalgebra h µ := g φ ∩ Der ( n ). These are the derivations which are traceless andcommute with φ , see paragraph following Theorem 8.2.If h µ is not reductive, then stop, n is not an Einstein nilradical.22f h µ is reductive then continue.To determine if this algebra is reductive: 1) compute its radical, then 2) compute the set of nilpotentelements of this radical. The algebra is reductive if and only if the set of such nilpotent elements (in theradical) is trivial. Step 3: Consider the subalgebra i g φ ( h µ ) = { X ∈ z g φ ( h µ ) | tr ( XY ) = 0 for all Y ∈ z g φ ( h µ ) ∩ h µ } (cf.Proposition 5.13), where z a ( b ) denotes the centralizer of b in a . Let D denote the matrices of gl n R which arediagonalizable over R ; i.e., D = [ g ∈ GL n R g t g − , where t = diagonal matrices of gl n R .Let n = n µ corresponding to some point µ ∈ V = ∧ ( R n ) ∗ ⊗ R n (see Section 3). For X ∈ i g ( h µ ) ∩ D ,write µ = P a i µ i , where µ i is an eigen basis for X , i.e., X · µ = P λ i a i µ i .If there is some X ∈ i g φ ( h µ ) ∩ D such that λ i ≥ a i = 0, then n is not an Einstein nilradical.If for every X above there exists i with λ i < a i = 0, then n is an Einstein nilradical. Remark 9.3. In Step 3,1) The identification of n with µ ∈ V is made by picking a basis of the vector space. This is tantamountto prescribing n with an orthonormal basis, and hence, endowing N with a choice of left-invariant metric.2) The X · µ , with X ∈ gl n R , precisely represent infinitesimal deformations of the above choice of left-invariant metric.3) The algebra h µ is reductive (once getting to Step 3). If the inner product from n µ makes h µ stableunder the metric adjoint (and there will always be such a µ with this property), then Step 3 may be replacedby the following. Step 3’: Assuming h µ is stable under the adjoint relative to the inner product on n µ , we may reduce thecollection of X considered in Step 3 to those X ∈ i g φ ( h µ ) ∩ p , where p = { Y ∈ h µ | Y t = Y } . Remark 9.4. The verification of Steps 1 and 2 above can done by a computer. It is not immediately clearto the author if Step 3 can be adapted to be implemented by a computer. Proof of the algorithm above Step 1: This is the content of Proposition 8.4. Step 2: To prove this portion of the algorithm, we will go ahead and identify n with n µ , for some µ ∈ V .The algebra h = g φ ∩ Der ( n ) is precisely the stabilizer subalgebra of g φ at µ . As we have fixed a basis ofour Lie algebra, we may view h ⊂ gl ( n, R ).In Theorem 8.3, it was shown that n µ is an Einstein nilradical if and only if G φ · µ is closed, where G φ is the (alegbraic) Lie group with Lie algebra g φ . It is well-known that an orbit being closed implies thestabilizer subgroup is reductive, see [RS90]. Lastly, the stabilizer subgroup is reductive if and only if its Liealgebra h is reductive. Step 3: As h and φ are reductive, there exists g ∈ GL ( n, R ) such that g h g − and gφg − are simultaneously θ -stable, i.e. closed under the transpose operation, see [Mos55]. Observe that gDer ( µ ) g − = Der ( g · µ ), gφg − is a pre-Einstein derivation of g · µ , g gφg − = g g φ g − , h g · µ = g h µ g − , z g gφg − ( h g · µ ) = g z g φ ( h µ ) g − , and i g gφg − ( h g · µ ) = g i g φ ( h µ ) g − G gφg − · ( g · µ ) = gG φ g − gµ = g ( G φ · µ ) is closed if and only if G φ · µ is closed. As such, we mayreduce to the case that h µ and φ are θ -stable.Since φ is θ -stable, we immediately have that g φ is θ -stable. Similarly, z g φ ( h µ ) is θ -stable. Now i g φ ( h µ )is precisely the Lie algebra of the algebraic reductive group I G φ ( H µ ) from Proposition 5.13, where H µ is theLie group with Lie algebra h µ .By Theorem 5.7, G φ · µ is closed if and only if Z G φ ( H µ ) is closed. And since I G φ ( H µ ) · µ = Z G φ ( H µ ) · µ ,we see that n µ is an Einstein nilradical if and only if I G φ ( H µ ) · µ is closed, see Proposition 5.13 and Theorem8.3.Observe that the stabilizer subalgebra of i g φ ( h µ ) at µ is trivial since it is contained in the stabilizer of g φ at µ (which equals h µ ) and i g φ ( h µ ) is orthogonal to h µ under the inner product h A, B i = tr ( AB t ). Hence,the stabilizer of I G φ ( H µ ) is finite (as it is discrete and algebraic).As the stabilizer of I G φ ( H µ ) at µ is finite, we may apply the ‘Hilbert-Mumford criterion’ to determinewhen I G φ ( H µ ) · µ is closed. This criterion was adapted to the real setting in [Bir71] which states (in oursetting) I G φ ( H µ ) · µ is closed if and only if [ t ∈ R exp ( tX ) · µ is closed for all X ∈ D ∩ i g φ ( h µ )Roughly speaking, this criterion says that closedness of an orbit is equivalent to closedness of the orbits ofall algebraic reductive 1-parameter subgroups.To finish, we write exp ( tX ) · µ = expt ( tX ) P a i µ i = P e tλ i a i µ i where µ i is the eigenvector of X above.This set is not closed if and only if for all i such that a i = 0, either all λ i ≥ λ i ≤ 0. Observe thatreplacing X with − X changes the sign of the eigenvalues above and this step is proven. Step 3’: Reducing the Hilber-Mumford criterion to this smaller set of symmetric elements of h µ is thecontent of [RS90]. 10 Algorithm to determine if a solvable Lie group admits a left-invariant Einstein metric In this section, we show that the existence of an Einstein metric on a solvable Lie group can be determinedby purely local data, as in the case of nilsolitons and nilpotent Lie groups. A similar algorithm can bewritten to test for the existence of solsoliton metrics. Theorem 10.1. Let S be a solvable Lie group with Lie algebra s . The existence of a left-invariant Einsteinmetric on S can be determined by analyzing the following: 1) adjoint action of s on itself, 2) the commutatorsubalgebra n = [ s , s ] , and 3) infinitesimal deformations of any initial metric on n . Flat Einstein metrics Here we prove Theorem 10.1 in the case that scalar curvature is zero (such a Lie algebra is necessarily uni-modular). This amounts to showing that the solvable Lie algebra in question has the rigid algebraic structuredescribed by Milnor, see Proposition 2.7. Note, this case does not require any infinitesimal deformations ofmetrics on n .Consider the adjoint action ad s ⊂ Der ( s ) on s . Compute the nilradical n of s , i.e., the set of nilpotentelements. Compute a Levi decomposition ad s = T + N , and let t ⊂ s be such that ad t = T and dim t = dim T . Lemma 10.2. If s admits a flat metric, then t is an abelian subalgebra, ad T has only purely imaginaryeigenvalues for T ∈ t , and dim t + dim n = dim s . Proving this lemma proves the theorem as verifying the conditions on t in the lemma amount to simplyanalyzing the adjoint representation of s , and any algebra of this type admits a flat metric by Proposition2.7. 24 roof of lemma. Assume s admits a flat Einstein metric. Decompose s = a + n where n is the nilradical and a is an abelian subalgebra such that ad A has only purely imaginary eigenvalues for A ∈ a , cf. Proposition2.7.Observe that ad s = ad a + ad n is a Levi-decomposition of ad s . Thus ad a and T = ad t are equalup to conjugation by Aut ( s ) as they are both maximal reductive subalgebras of ad s (conjugacy of suchsubalgebras is the main result of [Mos56]). As the relevant properties of a do not change after applying anautomorphism, we may assume ad a = ad t . Now, the elements of t differ from the elements of a by onlyelements of the center. Hence t has precisely the same properties of a and the lemma is proven. Negative Einstein metrics Here we prove Theorem 10.1 in the case that scalar curvature is negative (such a Lie algebra is necessarilynon-unimodular). Let S be the solvable group in question with Lie algebra s . Denote by n the commutatorsubalgebra [ s , s ] of s . (Note: when s admits an Einstein metric, this will be the full nilradical.) Step 1: If n is not an Einstein nilradical, then stop, S cannot admit a negative Einstein metric.If n is an Einstein nilradical, then continue.This step can be determined using the algorithm of Section 9. Step 2: Find a solution φ to tr ( φψ ) = tr ( ψ ) for all ψ ∈ Der ( n )within the set ad s = { ad X | X ∈ s } ⊂ Der ( n ).If there is no non-trivial solution in this subset, or the solution is not semisimple with (positive)real eigenvalues, then stop; S cannot admit a negative Einstein metric.If there is a non-trivial solution φ = ad X φ , and this solution is semisimple with (positive) realeigenvalues, then continue. Fix this choice of X φ .This step can be verified using a computer for a given solvable Lie algebra of interest. As before, positivityof the eigenvalues will follow if the remaining steps are valid. Step 3: Compute z s ( X φ ) = { Y ∈ s | [ Y, X φ ] = 0 } If z s ( X φ ) is not abelian or dim z s ( X φ ) + dim n < dim s , then stop, S does not admit a negativeEinstein metric.If z s ( X φ ) is abelian, and dim z s ( X φ ) + dim n = dim s , then continue.Recall, z s ( X φ ) is automatically reductive as X φ is reductive, and z s ( X φ ) being reductive abelian implies thatno element is nilpotent. This step may be verified using a computer. Step 4: If some element of z s ( X φ ) has only purely imaginary eigenvalues, then stop; S does not admit anegative Einstein metric.If no element of z s ( X φ ) has only purely imaginary eigenvalues, then S admits a negative Einsteinmetric. Proof of the algorithm above. Step 1: This fact is well-known, see [Lau07]. Step 2: This is the content of a theorem of Nikolayevsky, see Theorem 8.2, and [Lau10, Proposition 4.3]. Step 3: This follows immediately from [Lau10, Theorem 4.8]. Step 4: This is the content of Theorem 5.11. 25 Appendix: Closed orbits for general representations. The above work concerning the geometry of orbits holds in the more general framework of representationsof reductive groups. We state this result and provide only a sketch of the proof, as the proof is similar tothe above case. We do not know of this statement appearing in the literature before. Closed Orbits Theorem A.1. Let G be a real reductive algebraic group acting linearly and rationally on a vector space V . Determining whether an orbit G · v is closed in V can be determined using only data from the inducedrepresentation of the Lie algebra g at the point v ∈ V . In the following, we will only consider G which is semi-simple and use the Killing form B which is Ad ( G )-invariant and symmetric. More generally, for a reductive group, one could use any bilinear form B : g × g → g which is Ad ( G )-invariant, symmetric, and has the property that { X ∈ g | [ X, α ] = 0 and B ( X, α ) = 0 } is the Lie algebra of an algebraic group for any α ∈ g which is tangent to a reductive, algebraic 1-parametersubgroup. The Killing form satisfies this condition. Sketch of proof. We follow the same argument as in Section 9.The first requirement is that h = g v be reductive. Let z g ( h ) = { X ∈ g | [ X, h ] = 0 } denote the centralizerof h in g . As before, consider i g ( h ) = { X ∈ z g ( h ) | B ( X, Y ) = 0 for all Y ∈ z g ( h ) ∩ h } These subalgebras are the Lie algebras of algebraic groups Z G ( H ) and I G ( H ), respectively, where H = G v .The orbit G · v is closed if and only if Z G ( H ) · v = I G ( H ) · v is closed. As I G ( H ) has finite stabilizer and wemay apply the Hilbert-Mumford criterion.Let D denote the matrices of gl n R which are diagonalizable over R ; i.e., D = [ g ∈ GL n R g t g − , where t =diagonal matrices of gl n R . Given X ∈ i g ( h ) ∩ D , write v = P a i v i where { v i } is an eigenvector basis of V with X · v i = λ i v i . The Hilbert-Mumford criterion states: I G ( H ) · v is not closed if and only if there exists X ∈ i g ( h ) satisfying λ i ≥ i such that a i = 0.In this way, we see that determining the closedness of G · v reduces to analyzing the stabilizer subalgebra g v and the representation of g at v . References [AK75] D. V. Alekseevski˘ı and B. N. Kimel’fel’d, Structure of homogeneous Riemannian spaces with zero Riccicurvature , Functional Anal. Appl. (1975), no. 2, 97–102. MR MR0402650 (53 Einstein manifolds , Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the1987 edition. MR MR2371700 (2008k:53084)[Bir71] David Birkes, Orbits of linear algebraic groups , Ann. of Math. (2) (1971), 459–475. MR MR0296077 (45 Homogeneous Einstein metrics and simplicial complexes , J. Differential Geom. (2004),no. 1, 79–165. MR 2153482 (2006m:53065)[BWZ04] C. B¨ohm, M. Wang, and W. Ziller, A variational approach for compact homogeneous Einstein manifolds ,Geom. Funct. Anal. (2004), no. 4, 681–733.[CK04] Bennett Chow and Dan Knopf, The Ricci flow: an introduction , Mathematical Surveys and Monographs,vol. 110, American Mathematical Society, Providence, RI, 2004. MR MR2061425 (2005e:53101)[DM82] Isabel Dotti Miatello, Ricci curvature of left invariant metrics on solvable unimodular Lie groups , Math.Z. (1982), no. 2, 257–263. MR MR661702 (84a:53044) Ebe08] Patrick Eberlein, Riemannian 2-step nilmanifolds with prescribed Ricci tensor , Geometric and probabilisticstructures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 167–195.[EJ09] Patrick Eberlein and Michael Jablonski, Closed orbits of semisimple group actions and the real Hilbert-Mumford function , New developments in Lie theory and geometry, Contemp. Math., vol. 491, Amer. Math.Soc., Providence, RI, 2009, pp. 283–321. MR MR2537062[GW88] Carolyn S. Gordon and Edward N. Wilson, Isometry groups of Riemannian solvmanifolds , Trans. Amer.Math. Soc. (1988), no. 1, 245–269. MR MR936815 (89g:53073)[Heb98] Jens Heber, Noncompact homogeneous Einstein spaces , Invent. Math. (1998), no. 2, 279–352. MRMR1632782 (99d:53046)[HSS08] Peter Heinzner, Gerald W. Schwarz, and Henrik St¨otzel, Stratifications with respect to actions of realreductive groups , Compos. Math. (2008), no. 1, 163–185. MR MR2388560 (2009a:32030)[Jab08a] Michael Jablonski, Detecting orbits along subvarieties via the moment map , arXiv:0810.5697 [math.DG] –to appear in M¨unster Journal of Math (2008).[Jab08b] , Distinguished orbits of reductive groups , arXiv:0806.3721v1 [math.DG] (2008).[Jab09] , Moduli of Einstein and non-Einstein nilradicals , arXiv:0902.1698 [math.DG] (2009).[Jab10] , A natural Riemannian function on nilpotent lie groups , in progress (2010).[Jen69] Gary R. Jensen, Homogeneous Einstein spaces of dimension four , J. Differential Geometry (1969), 309–349. MR MR0261487 (41 The scalar curvature of left-invariant riemannian metrics , Indiana Univ. Math. J. (1971),1125–1144.[Kir84] Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry , Mathematical Notes31, Princeton University Press, Princeton, New Jersey, 1984.[KN78] G. Kempf and L. Ness, The length of vectors in representation spaces , Springer Lecture Notes 732 (Copen-hagen), Algebraic Geometry, Proceedings, 1978, pp. 233–244.[Lau] Jorge Lauret, Personal communication .[Lau01a] , Ricci soliton homogeneous nilmanifolds , Math. Ann. (2001), no. 4, 715–733. MR MR1825405(2002k:53083)[Lau01b] , Standard Einstein solvmanifolds as critical points , Q. J. Math. (2001), no. 4, 463–470. MRMR1874492 (2002j:53048)[Lau03] , On the moment map for the variety of Lie algebras , J. Funct. Anal. (2003), no. 2, 392–423.[Lau07] , Einstein solvmanifolds are standard , arXiv:math.DG/0703472 – to appear in Ann. of Math. (2007).[Lau08] , Einstein solvmanifolds and nilsolitons , arxiv:math.DG/0806.0035 (2008).[Lau10] , Ricci soliton solvmanifolds , arXiv:math.DG/1002.0384 – to appear in Crelle’s Journal (2010).[LW07] Jorge Lauret and Cynthia Will, Einstein solvmanifolds: Existence and non-existence questions ,arXiv:math/0602502v3 [math.DG] (2007).[Mar01] Alina Marian, On the real moment map , Math. Res. Lett. (2001), no. 5-6, 779–788. MR MR1879820(2003a:53123)[Mil76] John Milnor, Curvatures of left invariant metrics on Lie groups , Advances in Math. (1976), no. 3,293–329. MR MR0425012 (54 Self-adjoint groups , Ann. of Math. (2) (1955), 44–55. MR MR0069830 (16,1088a)[Mos56] , Fully reducible subgroups of algebraic groups , Amer. J. Math. (1956), 200–221. MR MR0092928(19,1181f)[Nik05] Yu. G. Nikonorov, Noncompact homogeneous Einstein 5-manifolds , Geom. Dedicata (2005), 107–143.MR MR2171301 (2006h:53037)[Nik08a] Y. Nikolayevsky, Einstein solvmanifolds and the pre-Einstein derivation , (arXiv:0802.2137) to appear inTrans. Amer. Math. Soc. (2008).[Nik08b] , Einstein solvmanifolds attached to two-step nilradicals , arXiv:0805.0646v1 [math.DG] (2008). Nik08c] Yuri Nikolayevsky, Einstein solvmanifolds with a simple Einstein derivation , Geom. Dedicata (2008),87–102. MR MR2413331 (2009f:53064)[Nik08d] , Einstein solvmanifolds with free nilradical , Ann. Global Anal. Geom. (2008), no. 1, 71–87. MRMR2369187 (2008m:53120)[NM84] Linda Ness and David Mumford, A stratification of the null cone via the moment map , American Journalof Mathematics (1984), no. 6, 1281–1329.[Pay10] Tracy L. Payne, The existence of soliton metrics for nilpotent Lie groups , Geom. Dedicata (2010),71–88. MR MR2600946[Rag72] M. S. Raghunathan, Discrete subgroups of Lie groups , Springer-Verlag, New York, 1972, Ergebnisse derMathematik und ihrer Grenzgebiete, Band 68. MR MR0507234 (58 Minimum vectors for real reductive algebraic groups , J. London Math.Soc. (1990), 409–429.[Sja98] Reyer Sjamaar, Convexity properties of the moment mapping re-examined , Adv. Math. (1998), no. 1,46–91. MR MR1645052 (2000a:53148)[Whi57] Hassler Whitney, Elementary structure of real algebraic varieties , The Annals of Mathematics (1957),no. 3, 545–556, 2nd Ser.[Wil03] C.E. Will, Rank-one einstein solvmanifolds of dimension 7 , Diff. Geom. Appl. (2003), 307–318.[Wil10] Cynthia Will, A curve of nilpotent Lie algebras which are not Einstein nilradicals , Monatsh. Math. (2010), no. 4, 425–437. MR MR2600907[WZ86] McKenzie Y. Wang and Wolfgang Ziller, Existence and nonexistence of homogeneous Einstein metrics ,Invent. Math. (1986), no. 1, 177–194. MR 830044 (87e:53081)(1986), no. 1, 177–194. MR 830044 (87e:53081)