Diagram involutions and homogeneous Ricci-flat metrics

Diego Conti; Federico A. Rossi; Viviana del Barco

REVIEW 4 major objections 4 minor 41 references

Diagram involutions and homogeneous Ricci-flat metrics

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Pith's one-line read A purely diagrammatic condition forces entire families of indefinite Ricci-flat metrics on nilpotent Lie groups

desk verdict Arrow-breaking involutions give a clean new construction of indefinite Ricci-flat metrics on nice nilpotent Lie groups, but the universal dimension ≤7 claims rest on asserted exhaustive checks. read the letter →

arxiv 1908.05975 v2 pith:VJUSLR5C submitted 2019-08-16 math.DG

Diego Conti , Viviana del Barco , Federico A. Rossi This is my paper · ORCID

classification math.DG MSC 22E2553C5053C2517B30

keywords Ricci-flatmetricsnilpotentLiegroupspseudo-Riemannianhomogeneousnicealgebrasdiagramsarrow-breakinginvolutionsparabolicnilradicalstwo-step

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that indefinite Ricci-flat metrics on nilpotent Lie groups can be produced by a purely combinatorial condition, with no curvature computation once the condition holds. On a nice nilpotent Lie algebra, an "arrow-breaking involution" of the associated diagram is an order-two permutation of the basis that never maps an arrow to an arrow; Proposition 2.3 proves every metric in the corresponding $\sigma$-diagonal family is Ricci-flat. Using this, the authors establish existence of such metrics on all nilpotent Lie groups of dimension at most 6, on all nice nilpotent Lie groups of dimension at most 7, and on every two-step nilpotent Lie group attached to a graph, and they construct infinite families from parabolic nilradicals of the split simple Lie groups $\mathrm{SL}(n)$, $\mathrm{SO}(p,q)$, and $\mathrm{Sp}(n,\mathbb{R})$. Most of the metrics are nonflat. The upshot is that for a large class of homogeneous spaces, an Einstein-metric existence question becomes a finite check on a directed graph.

What carries the argument

The load-bearing object is the arrow-breaking involution: an order-two permutation $\sigma$ of the nodes of a nice diagram such that whenever an arrow from $x$ to $z$ labeled $y$ exists, neither $\sigma(x)\to\sigma(z)$ labeled $\sigma(y)$ nor the corresponding reversed arrow appears. It is used together with the $\sigma$-diagonal metric (2), whose only nonzero inner products pair $e_i$ with $e_{\sigma(i)}$. Proposition 2.3 shows that under the arrow-breaking condition every such metric is Ricci-flat; Lemma 2.11 recasts the same condition as the absence of $\sigma$-invariant divisors of the diagram polynomials $P_\Delta$ and $Q_\Delta$, and the partial order on diagrams (Lemma 5.1) reduces existence to maximal diagrams.

What would settle it

Check the maximality and arrow-breaking assertions for one 7-dimensional algebra where the paper says 'it is easy to check' or 'a similar argument proves': for instance, verify by direct enumeration that 64321:5 has no arrow-breaking involution but satisfies the displayed Ricci-flat parameter equations. More broadly, a single nice nilpotent Lie algebra of dimension at most 7 not isomorphic to any algebra in the paper's tables, or one of the listed maximal algebras to which an extra arrow can be added without violating the nice-diagram rules, would invalidate the claimed universality of Theorem 5.6.

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Extended reading notes

Core claim

The central discovery is that Ricci-flatness of a $\sigma$-diagonal metric on a nice nilpotent Lie algebra is not an accident of structure constants but a consequence of the diagram alone. An arrow-breaking involution makes the metric orthogonal to both $\operatorname{ad}\mathfrak{g}$ and $d\mathfrak{g}^*$, forcing the Ricci tensor to vanish for every choice of parameters $g_i$. The paper then proves that the combinatorial condition is abundant: it holds whenever the center is large relative to the algebra, in particular for all two-step nilpotent Lie algebras attached to a graph; it can be verified through the polynomial criterion of Lemma 2.11; and it yields Theorem 5.6 and Corollary 5.9 after reducing to the finite list of maximal nice diagrams through dimension 7. The paper also writes down explicit arrow-breaking involutions for parabolic nilradicals in types $A_n$, $B_n$, $C_n$, and one $G_2$ example, producing infinite families of nonflat Ricci-flat nilmanifolds.

Load-bearing premise

The universal claims through dimension 7 rest on the external classifications of nilpotent and nice nilpotent Lie algebras being complete, and on the paper's hand-checked assertion that its maximality list is exhaustive; if a missing or misclassified example exists, the corresponding 'every' statement fails.

Editorial extensions

If this is right

All nilpotent Lie groups of dimension at most 6 admit indefinite Ricci-flat metrics; in dimension 6 the metric can be chosen nonflat for every nonabelian group.
All nice nilpotent Lie groups of dimension at most 7 admit Ricci-flat metrics, with nonflat choices except for the abelian case and two low-dimensional exceptions.
Every two-step nilpotent Lie group associated to a graph carries a Ricci-flat metric, by a center-dimension bound that guarantees an arrow-breaking involution.
Parabolic nilradicals in $\mathrm{SL}(n)$, $\mathrm{SO}(p,q)$, and $\mathrm{Sp}(n,\mathbb{R})$ give infinite families of Ricci-flat, generically nonflat nilmanifolds; rational choices of parameters give compact quotients in infinitely many diffeomorphism types.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The diagram-only nature of the condition suggests a direct computational test in dimension 8 and beyond: once a list of nice diagrams is available, the polynomial criterion of Lemma 2.11 can be checked by exhaustion without solving the full nilpotent classification.
Because the lone 6-dimensional algebra without a nice basis still carries the same kind of metric, the construction may extend beyond nice algebras; a formulation using only ordered bases and a compatibility condition could cover all nilpotent Lie algebras.
The flat subfamilies inside the Ricci-flat families, for instance $g_1=g_3$ in the 64321:4 example, indicate that each arrow-breaking involution typically yields a stratified family where flatness is a lower-dimensional condition on the metric parameters.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper in one line: it gives a genuine new combinatorial method for constructing indefinite Ricci-flat metrics on nice nilpotent Lie groups, and the main mechanism is airtight; what is not airtight is the completeness of the low-dimensional classification checks that turn the mechanism into universal statements.

The new idea is the arrow-breaking involution. Earlier work by the same authors used diagram automorphisms; here they use involutions that break every arrow they touch. The payoff is Proposition 2.3: if the involution is arrow-breaking, every sigma-diagonal metric is Ricci-flat. That is a real theorem with a short, readable proof, and it recovers and extends known examples. The polynomial criterion (Lemma 2.11) is a nice reformulation, and the large-center existence result in Section 3 is solid. The applications to graph two-step algebras and to parabolic nilradicals of type A, B, and C look legitimate; the root-system verification in Section 4 is detailed and convincing.

The soft spot is exactly where the stress-test points. Theorem 5.3 claims a complete list of maximal nice nilpotent Lie algebras of dimension at most 7. The 6-dimensional cases are argued; the 7-dimensional cases are dismissed with 'a similar argument proves'. That is a load-bearing exhaustive assertion. If a 7-dimensional maximal diagram were missing or misclassified, the universal statements Proposition 5.4 and Theorem 5.6 would not hold as stated. The same applies to the exceptional algebra 64321:5, handled by explicit Ricci-flat metrics only for selected parameter values, and to N6,1,4, where Ricci-flatness is asserted as 'easy computations'. None of this suggests the theorems are false; the authors are clearly competent and the checks are plausibly correct. But the preprint does not let a referee verify completeness without redoing the classification from [10] and [21].

My advice: send it to a serious referee. The core method deserves publication and advances the subject. Ask the referee to verify that Table 2 is complete, or to require the authors to supply the verification as ancillary data.

Referee Report

4 major / 4 minor

Summary. The paper introduces arrow-breaking involutions of nice diagrams and proves (Proposition 2.3) that any σ-diagonal metric (2) on a nice nilpotent Lie algebra with arrow-breaking σ is Ricci-flat. Lemma 2.11 translates the condition into coprimality of the polynomials PΔ and QΔ, making the condition checkable combinatorially, and Proposition 2.7 gives sufficient conditions for the resulting metrics to be nonflat. These tools are applied systematically: Proposition 3.1 proves existence of arrow-breaking involutions when the codimension r of the center satisfies r ≤ s + 3, yielding Corollary 3.5 for two-step nilpotent Lie algebras attached to graphs; Section 4 constructs arrow-breaking involutions for infinite families of parabolic nilradicals in types A_n, B_n, C_n, and G_2. The final section uses a maximality table (Table 2) to prove Theorem 5.6 (every nice nilpotent Lie algebra of dimension ≤ 7 admits a Ricci-flat metric) and Corollary 5.9 (every nonabelian 6-dimensional nilpotent Lie algebra admits a nonflat Ricci-flat metric).

Significance. If the low-dimensional verification is completed, the paper provides a genuinely systematic combinatorial construction of indefinite Ricci-flat metrics on large classes of nilpotent Lie groups. The algebraic core is clean and largely parameter-free: the arrow-breaking condition does not depend on structure constants (Remark 2.4), and Lemma 2.11 makes the condition checkable via polynomials. The nonflatness criteria in Proposition 2.7 are concrete, and the infinite families in Section 4 are explicit and constructive. The main risk is that the universal claims for dimensions ≤ 7 rest on hand-checked classification data that are not displayed in the manuscript.

major comments (4)

[Theorem 5.3 and Table 2] In the proof of Theorem 5.3, the maximality of the seven-dimensional entries in Table 2 is asserted with the sentence 'A similar argument proves the maximality of the 7-dimensional Lie algebras in the list', without displaying the case analysis. Since Lemma 5.1 and Proposition 5.4 reduce the dimension ≤ 7 claim to the completeness and correctness of Table 2, this is a load-bearing step. I ask that the 7D maximality check be supplied in full, either as a detailed case analysis in an appendix or as a machine-checkable electronic supplement.
[Proposition 5.4] The proof of Proposition 5.4 rules out arrow-breaking involutions for 64321:5 and then says that Table 2 provides an arrow-breaking involution for 'each of the other nice nilpotent Lie algebras of dimension ≤ 7'. What is not shown is the domination step: for every nice Lie algebra of dimension ≤ 7 not isomorphic to 64321:5, there is an entry of Table 2 dominating it. Because Table 2 lists only maximal algebras, the implication in Lemma 5.1 requires this enumeration. Please include a complete list of all nice Lie algebras in dimensions ≤ 7 together with the dominating maximal diagram, or an electronic script that reproduces the check.
[Corollary 5.9 and preceding paragraph] The claim that every nonabelian 6-dimensional nilpotent Lie algebra has a nonflat Ricci-flat metric depends on two external or asserted facts: the uniqueness of N6,1,4 as the only 6-dimensional nilpotent Lie algebra without a nice basis (quoted from [21]) and the statement that 'easy computations show' that the listed involutions give Ricci-flat metrics on N6,1,4. Both facts are load-bearing for the universal statement; please provide the actual computation of the Ricci tensor for N6,1,4 (or a precise reference where it appears) and a clear location for the uniqueness result in [21].
[Corollary 5.7 and Table 3] The list of algebras not covered by Table 2 is asserted without verification, and Table 3 contains an apparent error: the name 75421:6 appears twice with different presentations, namely (0,0,e12,e13,e23,e15+e24,e14+e26+e35) and (0,0,0,-e12,e14,e15+e24,e13+e26+e45). Since the exhaustiveness of this table is needed for Corollary 5.7, please correct the labels and provide a verifiable enumeration of the 17+1 algebras.

minor comments (4)

[Introduction, page 1] 'Aleksveesky conjecture' is a typo for 'Alekseevsky conjecture'.
[Section 3] Several displayed formulas contain '/integerdivide' artifacts, for example 'gk−2 /integerdividebk−2' in the proof of Proposition 3.1; this appears to be an unresolved LaTeX macro and should be rendered as proper quotient notation.
[Example 5.5] The phrase 'for g1 = ±g3' is ambiguous about which of the two displayed parameter conditions applies, and the final formula 'g3 = g2^2(g1^2 − g4^2)/(g1^2 g4)' should be checked for missing parentheses.
[Table 1] Entries such as '141 + 4 1/2 families' and '152 + 4 1/2 families' are hard to read; use a consistent notation for half-families, such as 141 + 4.5.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the arrow-breaking construction is an independent sufficient condition, and Ricci-flatness is verified from the curvature formula rather than assumed.

full rationale

The central step (Proposition 2.3) is not circular: 'arrow-breaking' (Definition 2.2) is a purely combinatorial condition on the nice diagram, while Ricci-flatness is then deduced from the Ricci formula (3) and shown to hold for every sigma-diagonal metric (2) with arbitrary nonzero parameters. No parameter is fitted to a Ricci-flat subset and then renamed a prediction; the condition is only sufficient, as the exceptional algebra 64321:5 in Proposition 5.4 has no arrow-breaking involution yet still admits a Ricci-flat metric in Example 5.5, showing the theorem is not equivalent to its input. The low-dimensional completeness statements do rely on the prior classifications [10] and [21] and on a compressed hand check in Theorem 5.3 and Table 2; those are external classification inputs (and, for [10], a published classification by the same authors), not results that already contain the Ricci-flat conclusion, so their use is a correctness or verification risk rather than a circular reduction. The self-citations [9], [10], [11] provide the Ricci formula, the nice-Lie-algebra classification, and earlier examples, but the paper does not cite any of them as if they established the arrow-breaking-to-Ricci-flat theorem. The infinite families for graph Lie algebras and parabolic nilradicals are likewise constructed by explicit involutions and checked by Propositions 2.3 and 2.7. Accordingly, no equation or fitted parameter makes a predicted quantity equal to its input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central construction introduces no new physical or algebraic entities: arrow-breaking involutions and σ-diagonal metrics are definitions, not postulates. No parameters are fitted; σ-diagonal metrics depend on arbitrary nonzero coefficients g_i, but Ricci-flatness holds for all choices, so the existence theorem requires no tuning. The external inputs are prior classifications and standard Ricci-curvature formulas.

assumptions (4)

standard math Formula (3) for the Ricci tensor of a nice Lie algebra, ric(v,w)=1/2⟨dv^b,dw^b⟩ - 1/2⟨ad v, ad w⟩, taken from Ref. [9].
Used in the proof of Proposition 2.3 to reduce Ricci-flatness to orthogonality of ad g and d g*.
domain assumption Completeness of the classification of nice nilpotent Lie algebras up to dimension 7 in Ref. [10] and of nilpotent Lie algebras of dimension 6 in Ref. [21].
Theorem 5.3 and Corollary 5.9 rely on these classifications to assert universality; if the lists are incomplete, the existence theorems may miss algebras.
domain assumption The list of maximal nice nilpotent Lie algebras in Table 2 is complete, as asserted in Theorem 5.3.
Lemma 5.1 reduces existence to maximal diagrams; the proof proceeds 'Going through the classification of [10]' with several 'easy to check' steps.
standard math Nice bases and nice diagrams as defined in [10] encode the relevant Lie brackets, with nilpotency giving acyclicity.
Background framework for the paper; no novel postulate is introduced.

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Cite this review

Pith. "Pith review of Diagram involutions and homogeneous Ricci-flat metrics." pith.science (2026). https://pith.science/paper/VJUSLR5C

@misc{pith2026190805975,
  author       = {Pith},
  title        = {Pith review of: Diagram involutions and homogeneous Ricci-flat metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJUSLR5C}},
  note         = {Machine review of arXiv:1908.05975}
}

abstract

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.

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Works this paper leans on

41 extracted references · 40 canonical work pages

[10]

Conti and F

D. Conti and F. A. Rossi. Construction of nice nilpotent Lie grou ps. Journal of Algebra, 525:311 – 340, 2019

work page 2019
[21]

M.-P. Gong. Classiﬁcation of nilpotent Lie algebras of dimension 7 (ove r algebraically closed ﬁelds and R) . ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of Waterloo (Canada)

work page 1998
[1]

D. V. Alekseevsky and B. N. Kimel ′fel′d. Structure of homogeneous Rie- mannian spaces with zero Ricci curvature. Funkcional. Anal. i Prilo ˇZen., 9(2):5–11, 1975. 29

work page 1975
[2]

D. V. Alekseevsky, C. Medori, and A. Tomassini. Homogeneous pa ra- K¨ ahlerian Einstein manifolds.Uspekhi Mat. Nauk , 64(1(385)):3–50, 2009

work page 2009
[3]

Aubert and A

A. Aubert and A. Medina. Groupes de Lie pseudo-Riemanniens plat s. Tohoku Math. J. (2) , 55(4):487–506, 2003

work page 2003
[4]

A. L. Besse. Einstein manifolds . Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition

work page 2008
[5]

Bokan, T

N. Bokan, T. ˇSukilovi´ c, and S. Vukmirovi´ c. Lorentz geometry of 4- dimensional nilpotent Lie groups. Geom. Dedicata, 177:83–102, 2015

work page 2015
[6]

Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups

M. Boucetta. Ricci ﬂat left invariant Lorentzian metrics on 2-st ep nilpotent Lie groups. arXiv:0910.2563v2

work page Pith review arXiv

Show all 41 references

[7]

Calvaruso and A

G. Calvaruso and A. Zaeim. Four-dimensional Lorentzian Lie grou ps. Dif- ferential Geom. Appl. , 31(4):496–509, 2013

2013
[8]

Calvaruso and A

G. Calvaruso and A. Zaeim. Neutral metrics on four-dimensional Lie groups. J. Lie Theory , 25(4):1023–1044, 2015

2015
[9]

Conti and F

D. Conti and F. A. Rossi. Indeﬁnite Einstein metrics on nice Lie gro ups. arXiv:1805.08491

arXiv
[11]

Conti and F

D. Conti and F. A. Rossi. Ricci-ﬂat and Einstein pseudoriemannia n nil- manifolds. Complex Manifolds , 6(1):170–193, 2019

2019
[12]

S. G. Dani and M. G. Mainkar. Anosov automorphisms on compac t nilman- ifolds associated with graphs. Trans. Amer. Math. Soc. , 357(6):2235–2251, 2005

2005
[13]

del Barco and G

V. del Barco and G. P. Ovando. Free nilpotent Lie algebras admit ting ad-invariant metrics. J. Algebra, 366:205–216, 2012

2012
[14]

Derdzinski

A. Derdzinski. Curvature-homogeneous indeﬁnite Einstein metrics in di- mension four: the diagonalizable case , volume 337 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2003

2003
[15]

Derdzinski and ´S

A. Derdzinski and ´S. R. Gal. Indeﬁnite Einstein metrics on simple Lie groups. Indiana Univ. Math. J. , 63(1):165–212, 2014

2014
[16]

Der´ e and J

J. Der´ e and J. Lauret. On Ricci negative solvmanifolds and their nilradicals. Math. Nachr. , 292(7):1462–1481, 2019

2019
[17]

Favre and L

G. Favre and L. J. Santharoubane. Symmetric, invariant, non degenerate bilinear form on a Lie algebra. J. Algebra, 105(2):451–464, 1987

1987
[18]

Fino and I

A. Fino and I. Kath. Holonomy groups of G∗ 2-manifolds. Trans. Amer. Math. Soc. , 371(11):7725–7755, 2019

2019
[19]

Fino and I

A. Fino and I. Luj´ an. Torsion-free G∗ 2(2)-structures with full holonomy on nilmanifolds. Adv. Geom., 15(3):381–392, 2015. 30

2015
[20]

Freibert

M. Freibert. Calibrated and parallel structures on almost Abelia n Lie al- gebras. arXiv:1307.2542

arXiv
[22]

M. Guediri. Lorentz Geometry of 2-Step Nilpotent Lie Groups. Geom. Dedicata, 100(1):11–51, 2003

2003
[23]

Guediri and M

M. Guediri and M. Bin-Asfour. Ricci-ﬂat left-invariant Lorentz ian metrics on 2-step nilpotent Lie groups. Arch. Math. (Brno) , 50(3):171–192, 2014

2014
[24]

J. Heber. Noncompact homogeneous Einstein spaces. Invent. Math. , 133(2):279–352, 1998

1998
[25]

Ivanov and S

S. Ivanov and S. Zamkovoy. Parahermitian and paraquaternio nic manifolds. Diﬀerential Geom. Appl. , 23(2):205–234, 2005

2005
[26]

G. R. Jensen. The scalar curvature of left-invariant Riemannia n metrics. Indiana Univ. Math. J. , 20:1125–1144, 1970/1971

1970
[27]

I. Kath. Pseudo-Riemannian T -duals of compact Riemannian homogeneous spaces. Transform. Groups, 5(2):157–179, 2000

2000
[28]

I. Kath. Indeﬁnite symmetric spaces with G 2(2)-structure. J. Lond. Math. Soc. (2) , 87(3):853–876, 2013

2013
[29]

B. Kostant. Root systems for Levi factors and Borel-de Sieb enthal the- ory. In Symmetry and spaces , volume 278 of Progr. Math., pages 129–152. Birkh¨ auser Boston, Inc., Boston, MA, 2010

2010
[30]

J. Lauret. Einstein solvmanifolds are standard. Ann. of Math. (2) , 172(3):1859–1877, 2010

2010
[31]

Lauret and C

J. Lauret and C. Will. Einstein solvmanifolds: existence and non-e xistence questions. Math. Ann. , 350(1):199–225, 2011

2011
[32]

Lauret and C

J. Lauret and C. Will. On the diagonalization of the Ricci ﬂow on Lie groups. Proc. Amer. Math. Soc. , 141(10):3651–3663, 2013

2013
[33]

L. Magnin. Sur les alg` ebres de Lie nilpotentes de dimension ≤ 7. J. Geom. Phys., 3(1):119–144, 1986

1986
[34]

A. I. Malcev. On a class of homogeneous spaces. Amer. Math. Soc. Trans- lation, 1951(39):33, 1951

1951
[35]

B. O’Neill. Semi-Riemannian geometry , volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publish- ers], New York, 1983. With applications to relativity

1983
[36]

G. P. Ovando. Lie algebras with ad-invariant metrics: A survey- guide. Rend. Semin. Mat. Univ. Politec. Torino , 74(1):243–268, 2016. 31

2016
[37]

Sch¨ afer and F

L. Sch¨ afer and F. Schulte-Hengesbach. Nearly pseudo-K¨ a hler and nearly para-K¨ ahler six-manifolds. In Handbook of pseudo-Riemannian geometry and supersymmetry , volume 16 of IRMA Lect. Math. Theor. Phys. , pages 425–453. Eur. Math. Soc., Z¨ urich, 2010

2010
[38]

ˇSukilovi´ c

T. ˇSukilovi´ c. Geometric properties of neutral signature metrics on 4- dimensional nilpotent Lie groups. Rev. Uni´ on Mat. Argent., 57(1):23–47, 2016

2016
[39]

H. Tamaru. Parabolic subgroups of semisimple Lie groups and Eins tein solvmanifolds. Math. Ann. , 351(1):51–66, 2011

2011
[40]

M. Y. Wang and W. Ziller. Existence and nonexistence of homogen eous Einstein metrics. Invent. Math. , 84(1):177–194, 1986

1986
[41]

J. A. Wolf. The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. , 120:59–148, 1968. Dipartimento di Matematica e Applicazioni, Universit` a di Milano Bicocca, via Cozzi 55, 20125 Milano, Italy. diego.conti@unimib.it federico.rossi@unimib.it Univer...

1968

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Conti and F

D. Conti and F. A. Rossi. Construction of nice nilpotent Lie grou ps. Journal of Algebra, 525:311 – 340, 2019

work page 2019

[2] [21]

M.-P. Gong. Classiﬁcation of nilpotent Lie algebras of dimension 7 (ove r algebraically closed ﬁelds and R) . ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of Waterloo (Canada)

work page 1998

[3] [1]

D. V. Alekseevsky and B. N. Kimel ′fel′d. Structure of homogeneous Rie- mannian spaces with zero Ricci curvature. Funkcional. Anal. i Prilo ˇZen., 9(2):5–11, 1975. 29

work page 1975

[4] [2]

D. V. Alekseevsky, C. Medori, and A. Tomassini. Homogeneous pa ra- K¨ ahlerian Einstein manifolds.Uspekhi Mat. Nauk , 64(1(385)):3–50, 2009

work page 2009

[5] [3]

Aubert and A

A. Aubert and A. Medina. Groupes de Lie pseudo-Riemanniens plat s. Tohoku Math. J. (2) , 55(4):487–506, 2003

work page 2003

[6] [4]

A. L. Besse. Einstein manifolds . Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition

work page 2008

[7] [5]

Bokan, T

N. Bokan, T. ˇSukilovi´ c, and S. Vukmirovi´ c. Lorentz geometry of 4- dimensional nilpotent Lie groups. Geom. Dedicata, 177:83–102, 2015

work page 2015

[8] [6]

Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups

M. Boucetta. Ricci ﬂat left invariant Lorentzian metrics on 2-st ep nilpotent Lie groups. arXiv:0910.2563v2

work page Pith review arXiv