scholarly journals Homogeneous Ricci solitons

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Michael Jablonski
Keyword(s):  
Lie Groups ◽  
Ricci Flow ◽  
Isometry Group ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Solvable Lie Groups ◽  
Compact Case ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Special Case

AbstractIn this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝIn the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.

scholarly journals Linear stability of algebraic Ricci solitons

2016 ◽  
Vol 2016 (713) ◽  
Author(s):  
Michael Jablonski ◽  
Peter Petersen ◽  
Michael Bradford Williams
Keyword(s):  
Linear Stability ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Ricci Soliton ◽  
Stationary Solutions ◽  
Flow Equation ◽  
Solvable Lie Groups ◽  
Codimension One ◽  
Open Set ◽  
Simply Connected

AbstractWe consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all solvable Einstein metrics of dimension seven or less with codimension-one nilradical. For each linearly stable metric, dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.

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 The Guillemin–Sternberg conjecture for noncompact groups and spaces

2008 ◽  
Vol 1 (3) ◽  
pp. 473-533 ◽  
Author(s):  
P. Hochs ◽  
N.P. Landsman
Keyword(s):  
Normal Subgroup ◽  
Dirac Operator ◽  
Lie Groups ◽  
Group Action ◽  
Symplectic Manifolds ◽  
Lie Group Action ◽  
Compact Case ◽  
Assembly Map ◽  
High Degree ◽  
Special Case

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

scholarly journals On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

2021 ◽  
pp. 86-90
Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Natural Generalization ◽  
Ricci Solitons ◽  
Tensor Fields ◽  
Left Invariant Riemannian Metric ◽  
Riemannian Case

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

Restricting Representations of Completely Solvable Lie Groups

1990 ◽  
Vol 42 (5) ◽  
pp. 790-824 ◽  
Author(s):  
R. L. Lipsman
Keyword(s):  
Lie Groups ◽  
Dual Problem ◽  
Nilpotent Groups ◽  
Solvable Lie Groups ◽  
Direct Integral ◽  
Induced Representations ◽  
Simply Connected ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Restriction Problem

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

Pseudo-algebraic Ricci solitons on Einstein nilradicals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zaili Yan
Keyword(s):  
Variational Method ◽  
Lie Groups ◽  
Lie Group ◽  
Ricci Soliton ◽  
Nilpotent Lie Groups ◽  
Ricci Solitons ◽  
Codimension One ◽  
Abelian Ideal ◽  
Einstein Nilradical

Abstract We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups. As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

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