Ricci curvature of left invariant metrics on solvable unimodular Lie groups

1982 ◽  
Vol 180 (3) ◽  
pp. 257-263 ◽  
Author(s):  
Isabel Dotti Miatello
Keyword(s):  
Lie Groups ◽  
Ricci Curvature ◽  
Invariant Metrics ◽  
Left Invariant Metrics

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 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).

Completeness of left-invariant metrics on Lie groups

1987 ◽  
Vol 21 (3) ◽  
pp. 233-234
Author(s):  
D. V. Alekseevskii ◽  
B. A. Putko
Keyword(s):  
Lie Groups ◽  
Invariant Metrics ◽  
Left Invariant Metrics

scholarly journals Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

2019 ◽  
Vol 31 (6) ◽  
pp. 1567-1605 ◽  
Author(s):  
Gabriel Larotonda
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Linear Operators ◽  
Finsler Metric ◽  
Invariant Metrics ◽  
Metric Geometry ◽  
Infinite Dimensional ◽  
Left Invariant Metrics ◽  
Geodesic Structure ◽  
Infinite Dimensional Lie Groups

AbstractWe study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

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.

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