scholarly journals The Ricci pinching functional on solvmanifolds

2019 ◽  
Author(s):  
Jorge Lauret ◽  
Cynthia E Will
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Invariant Metrics ◽  
Solvable Lie Groups ◽  
Left Invariant Metrics ◽  
Solvable Lie Group

Abstract We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

scholarly journals Classification of left invariant metrics on 4-dimensional solvable Lie groups

2020 ◽  
pp. 14-14
Author(s):  
Tijana Sukilovic
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Vector Fields ◽  
Classification Problem ◽  
Inner Product ◽  
Invariant Metrics ◽  
Solvable Lie Groups ◽  
Left Invariant Metrics

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Topology on the unitary dual of completely solvable Lie groups

2015 ◽  
Vol 6 (4) ◽  
Author(s):  
Detlev Poguntke
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Orbit Space ◽  
Orbit Method ◽  
Solvable Lie Groups ◽  
Bijective Correspondence ◽  
Unitary Dual ◽  
Exponential Lie Group ◽  
Solvable Lie Group

AbstractIt was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some “philosophy” and some new arguments are added. The purpose is to contribute to a better understanding of this proof.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

Exponentiality of Certain Real Solvable Lie Groups

1998 ◽  
Vol 41 (3) ◽  
pp. 368-373 ◽  
Author(s):  
Martin Moskowitz ◽  
Michael Wüstner
Keyword(s):  
Lie Groups ◽  
Semidirect Product ◽  
Lie Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Solvable Lie Groups ◽  
Necessary And Sufficient ◽  
Solvable Lie Group

AbstractIn this article, making use of the second author’s criterion for exponentiality of a connected solvable Lie group, we give a rather simple necessary and sufficient condition for the semidirect product of a torus acting on certain connected solvable Lie groups to be exponential.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Singularity of the varieties of representations of lattices in solvable Lie groups

2016 ◽  
Vol 08 (02) ◽  
pp. 273-285 ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Vector Bundles ◽  
Similar Technique ◽  
Trivial Representation ◽  
Nilpotent Lie Algebra ◽  
Solvable Lie Groups ◽  
Analytic Germ ◽  
Space Construction ◽  
Simply Connected ◽  
Solvable Lie Group

For a lattice [Formula: see text] of a simply connected solvable Lie group [Formula: see text], we describe the analytic germ in the variety of representations of [Formula: see text] at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by [Formula: see text]. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of [Formula: see text] at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.

Naturally Reductive Homogeneous Riemannian Manifolds

1985 ◽  
Vol 37 (3) ◽  
pp. 467-487 ◽  
Author(s):  
Carolyn S. Gordon
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Invariant Metrics ◽  
List Type ◽  
Left Invariant Metrics ◽  
Homogeneous Riemannian Manifolds ◽  
Transitive Subgroup ◽  
Naturally Reductive

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

scholarly journals Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

2007 ◽  
Vol 50 (1) ◽  
pp. 24-34 ◽  
Author(s):  
Nathan Brown ◽  
Rachel Finck ◽  
Matthew Spencer ◽  
Kristopher Tapp ◽  
Zhongtao Wu
Keyword(s):  
Sectional Curvature ◽  
Lie Groups ◽  
Nonnegative Curvature ◽  
Invariant Metrics ◽  
Compact Lie Groups ◽  
Nonnegative Sectional Curvature ◽  
Left Invariant Metrics

AbstractWe classify the left-invariant metrics with nonnegative sectional curvature on SO(3) and U(2).

Sign in / Sign up

Export Citation Format

Share Document