Classification of invariant Einstein metrics on certain compact homogeneous spaces

2019 ◽  
Vol 63 (4) ◽  
pp. 755-776
Author(s):  
Zaili Yan ◽  
Huibin Chen ◽  
Shaoqiang Deng
Keyword(s):  
Einstein Metrics ◽  
Homogeneous Spaces

Invariant K�hler?Einstein metrics on compact homogeneous spaces

1986 ◽  
Vol 20 (3) ◽  
pp. 171-182 ◽  
Author(s):  
D. V. Alekseevskii ◽  
A. M. Perelomov
Keyword(s):  
Einstein Metrics ◽  
Homogeneous Spaces

scholarly journals Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups

2009 ◽  
Vol 61 (6) ◽  
pp. 1201-1213 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
V. V. Dzhepko ◽  
Yu. G. Nikonorov
Keyword(s):  
Riemannian Manifold ◽  
Positive Integer ◽  
Lie Groups ◽  
Einstein Metrics ◽  
Homogeneous Spaces ◽  
Classical Groups ◽  
Stiefel Manifolds ◽  
Additional Symmetries

Abstract A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds SO(n)/SO(l). Furthermore, we show that for any positive integer p there exists a Stiefelmanifold SO(n)/SO(l) that admits at least p SO(n)-invariant Einstein metrics.

Ambitoric geometry I: Einstein metrics and extremal ambikähler structures

2016 ◽  
Vol 2016 (721) ◽  
Author(s):  
Vestislav Apostolov ◽  
David M. J. Calderbank ◽  
Paul Gauduchon
Keyword(s):  
Maxwell Equations ◽  
Einstein Metrics ◽  
Natural Extension ◽  
Torus Action ◽  
Local Geometry ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Bach Tensor ◽  
Extremal Kähler Metrics

AbstractWe present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kähler metrics which are toric with respect to a common 2-torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomialWe use this description to classify 4-dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors.These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures,

Affine compact almost-homogeneous manifolds of cohomogeneity one

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Daniel Guan
Keyword(s):  
Einstein Metrics ◽  
Compact Manifolds ◽  
Complex Manifolds ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Kähler Einstein Metrics

AbstractThis paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

Classification of five-dimensional naturally reductive spaces

1985 ◽  
Vol 97 (3) ◽  
pp. 445-463 ◽  
Author(s):  
Oldřich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
General Theory ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Natural Generalization ◽  
Riemannian Symmetric Spaces ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive ◽  
Volume Preserving ◽  
Number Of Authors

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

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