On solvable Lie groups of negative Ricci curvature

Nikolayevsky, Yuri; Nikonorov, Yu. G.

doi:10.1007/s00209-015-1410-2

Cited by 11 publications

(35 citation statements)

References 18 publications

(7 reference statements)

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“…It is proved there that if V is a non-trivial real representation of su(2) extended to u(2) by letting the center act as multiples of the identity, then the Lie algebra u(2) ⋉ V admits an inner product with negative Ricci curvature. Before that, the only Lie groups in the literature that were known to admit a left-invariant metric with negative Ricci curvature were either semisimple (see [3], [4]) or solvable (see [2], [16], [17], [13]). We refer to [16], [19] or [13] for a more detailed summary of the known results on negative Ricci curvature in the homogeneous case.…”

Section: Introductionmentioning

confidence: 99%

Negative Ricci curvature on some non-solvable Lie groups II

Will

2019

Math. Z.

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We construct many examples of Lie groups admitting a left-invariant metric of negative Ricci curvature. We study Lie algebras which are semidirect products l = (a ⊕ u) ⋉ n and we obtain examples where u is any semisimple compact real Lie algebra, a is one-dimensional and n is a representation of u which satisfies some conditions. In particular, when u = su(m), so(m) or sp(m) and n is a representation of u in some space of homogeneous polynomials, we show that these conditions are indeed satisfied. In the case u = su(2) we get a more general construction where n can be any nilpotent Lie algebra where su(2) acts by derivations. We also prove a general result in the case when u is a semisimple Lie algebra of non-compact type.

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Section: Introductionmentioning

confidence: 99%

Negative Ricci curvature on some non-solvable Lie groups II

Will

2019

Math. Z.

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“…In this paper we continue the study of metric solvable Lie groups admitting a left‐invariant metric of negative Ricci curvature, which has been started in , and we refer the reader to the Introduction of that paper for a detailed overview of known results. At present, necessary and sufficient conditions for a homogeneous space to admit a left‐invariant metric with a particular sign of the sectional curvature are well understood, as well as the conditions for a homogeneous space to admit a left‐invariant metric with positive or with zero Ricci curvature.…”

Section: Introductionmentioning

confidence: 99%

“…The precise nature of such inequalities depends on the structure of

frakturn

and in the general case remains unknown. In the authors speculated that they may be related to the fact that (the real semisimple part of) the derivation belongs to a certain convex cone in the torus of derivations of

frakturn

.…”

Section: Introductionmentioning

confidence: 99%

“…The necessary and sufficient conditions to admit an inner product with

Ric < 0

for Lie algebras whose nilradical is a Heisenberg Lie algebra or is a standard filiform algebra have a similar “flavour” .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem Let

frakturg

be a solvable non‐nilpotent Lie algebra with the filiform nilradical

frakturn

. Then

frakturg

admits an inner product of negative Ricci curvature if and only if (a)either

n ≃ L_{n}

and

frakturg

is a one‐dimensional extension of

frakturn

by a vector Y such that

ι_{2} (Y), ι_{n} (Y) > 0

[ , Theorem 4 ] ; (b)or

n ≃ Q_{n}, n = 2 m, m > 2

, and

frakturg

is a one‐dimensional extension of

frakturn

by a vector Y such that

ι_{k} (Y) > 0

, for all

k = m + 1, \dots, n

; (c)or, with no restrictions, in all the other cases: either

frakturn

is any other filiform algebra, or otherwise

n ≃ L_{n}

n ≃ Q_{n} (n = 2 m > 4)

, and

frakturg

is an extension of

frakturn

of dimension greater than one. …”

Section: Introductionmentioning

confidence: 99%

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Solvable extensions of negative Ricci curvature of filiform Lie groups

Nikolayevsky

2015

Mathematische Nachrichten

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Abstract. We give necessary and sufficient conditions of the existence of a leftinvariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra g is a filiform Lie algebra n. It turns out that such a metric always exists, except for in the two cases, when n is one of the algebras of rank two, L n or Q n , and g is a one-dimensional extension of n, in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.

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On Ricci negative solvmanifolds and their nilradicals

Deré

Lauret

2019

Mathematische Nachrichten

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In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications. K E Y W O R D S negative, Ricci, solvmanifold M S C ( 2 0 1 0 ) 53C20, 53C30 1462

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On solvable Lie groups of negative Ricci curvature

Cited by 11 publications

References 18 publications

Negative Ricci curvature on some non-solvable Lie groups II

Negative Ricci curvature on some non-solvable Lie groups II

Solvable extensions of negative Ricci curvature of filiform Lie groups

On Ricci negative solvmanifolds and their nilradicals

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