On naturally reductive left-invariant metrics of SL (2, R)

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.

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

International Journal of Mathematics ◽

10.1142/s0129167x18500830 ◽

2018 ◽

Vol 29 (11) ◽

pp. 1850083 ◽

Cited By ~ 1

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.

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The Ricci pinching functional on solvmanifolds

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haz020 ◽

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.

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A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

Journal of Symbolic Logic ◽

10.2178/jsl/1164060447 ◽

2006 ◽

Vol 71 (4) ◽

pp. 1108-1124 ◽

Cited By ~ 1

Author(s):

Alex Thompson

Keyword(s):

Equivalence Relation ◽

Polish Space ◽

Invariant Metrics ◽

Orbit Equivalence ◽

Invariant Metric ◽

Continuous Action ◽

Polish Groups ◽

Polish Group ◽

Left Invariant Metrics

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

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