Naturally reductive metrics and Einstein metrics on compact Lie groups

1979 ◽  
Vol 18 (215) ◽  
pp. 0-0 ◽  
Author(s):  
J. E. D’Atri ◽  
W. Ziller
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Compact Lie Groups ◽  
Naturally Reductive ◽  
Naturally Reductive Metrics

scholarly journals Einstein metrics on compact Lie groups which are not naturally reductive

2011 ◽  
Vol 160 (1) ◽  
pp. 261-285 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Kunihiko Mori ◽  
Yusuke Sakane
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Compact Lie Groups ◽  
Naturally Reductive

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.

Left Invariant Einstein–Randers Metrics on Compact Lie Groups

2012 ◽  
Vol 55 (4) ◽  
pp. 870-881 ◽  
Author(s):  
Hui Wang ◽  
Shaoqiang Deng
Keyword(s):  
Lie Groups ◽  
Complete Classification ◽  
Simple Groups ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Navigation Data ◽  
Group Of Isometries ◽  
Randers Metrics ◽  
Naturally Reductive

AbstractIn this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

scholarly journals Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

2020 ◽  
pp. 2050079
Author(s):  
Zaili Yan ◽  
Shaoqiang Deng
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Complete Classification ◽  
Semisimple Lie Groups ◽  
Killing Form ◽  
Interesting Phenomenon ◽  
Product Metrics ◽  
Coset Spaces ◽  
Naturally Reductive

A quadruple of Lie groups [Formula: see text], where [Formula: see text] is a compact semisimple Lie group, [Formula: see text] are closed subgroups of [Formula: see text], and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces [Formula: see text], [Formula: see text] and [Formula: see text] are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple [Formula: see text] in our list, we can produce new non-naturally reductive Einstein metrics on the coset space [Formula: see text], by scaling the Killing form metrics along the complement of [Formula: see text] in [Formula: see text] and along the complement of [Formula: see text] in [Formula: see text]. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

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