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).

New non-naturally reductive Einstein metrics on SO(n)

2018 ◽  
Vol 29 (11) ◽  
pp. 1850083 ◽  
Author(s):  
Bo Zhang ◽  
Huibin Chen ◽  
Ju Tan
Keyword(s):  
Lie Groups ◽  
Ricci Tensor ◽  
Einstein Metrics ◽  
Gröbner Bases ◽  
Invariant Metrics ◽  
Polynomial Equations ◽  
Compact Lie Groups ◽  
Symbolic Computations ◽  
Left Invariant Metrics ◽  
Naturally Reductive

We obtain new invariant Einstein metrics on the compact Lie groups [Formula: see text] ([Formula: see text]) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on [Formula: see text] and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

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 Spectral isolation of bi-invariant metrics on compact Lie groups

2010 ◽  
Vol 60 (5) ◽  
pp. 1617-1628 ◽  
Author(s):  
Carolyn S. Gordon ◽  
Dorothee Schueth ◽  
Craig J. Sutton
Keyword(s):  
Lie Groups ◽  
Invariant Metrics ◽  
Compact Lie Groups

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.

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