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Einstein metrics on aligned homogeneous spaces with two factors

Jorge Lauret, Cynthia Will

TL;DR

The paper studies G-invariant Einstein metrics on aligned spaces M = G1 × G2 / K, where G1 and G2 are compact simple groups and K projects nontrivially onto both factors. It shows that, for the large subclass with three isotropy summands, Einstein existence for metrics of the form g = (x1,x2,x3) is equivalent to the real-root condition of a quartic polynomial p whose coefficients depend on dimensions and two Killing constants, enabling a complete classification within this subclass. Focusing on the class C (K simple, two isotropy irreducibles), the authors compute discriminants and related invariants to classify 12 infinite families and 70 sporadic examples, finding an overall existence rate of about 75% and exactly two invariant Einstein metrics in the existence cases. The work extends prior two-component isotropy results to aligned spaces with two simple factors, providing a practical polynomial criterion and a detailed catalog of new examples and non-existence instances. The results have implications for understanding stability and topology in homogeneous Einstein geometry and for constructing explicit examples in the aligned two-factor setting.

Abstract

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.

Einstein metrics on aligned homogeneous spaces with two factors

TL;DR

The paper studies G-invariant Einstein metrics on aligned spaces M = G1 × G2 / K, where G1 and G2 are compact simple groups and K projects nontrivially onto both factors. It shows that, for the large subclass with three isotropy summands, Einstein existence for metrics of the form g = (x1,x2,x3) is equivalent to the real-root condition of a quartic polynomial p whose coefficients depend on dimensions and two Killing constants, enabling a complete classification within this subclass. Focusing on the class C (K simple, two isotropy irreducibles), the authors compute discriminants and related invariants to classify 12 infinite families and 70 sporadic examples, finding an overall existence rate of about 75% and exactly two invariant Einstein metrics in the existence cases. The work extends prior two-component isotropy results to aligned spaces with two simple factors, providing a practical polynomial criterion and a detailed catalog of new examples and non-existence instances. The results have implications for understanding stability and topology in homogeneous Einstein geometry and for constructing explicit examples in the aligned two-factor setting.

Abstract

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.
Paper Structure (11 sections, 9 theorems, 91 equations, 2 figures, 5 tables)

This paper contains 11 sections, 9 theorems, 91 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Aligned homogeneous spaces
  3. Definition
  4. Examples
  5. Reductive decomposition
  6. Ricci curvature
  7. Structural constants
  8. Einstein metrics
  9. $K$ abelian
  10. $K$ semisimple
  11. The class $\mathcal{C}$

Key Result

Theorem 1.1

If an aligned homogeneous space $M=G_1\times G_2/K$ admits an Einstein metric of the form $g=(x_1,x_2,x_3)$, then, for $i=1,2$, the Casimir operator of $G_i/\pi_i(K)$ is given by $\operatorname{C}_{\chi_i}=\kappa_iI_{\mathfrak{p} _i}$ for some $\kappa_i>0$ (i.e., the standard metric on $G_i/\pi_i(K) Moreover, the Einstein metric $g$ is always unstable as a critical point of the scalar curvature fu

Figures (2)

  • Figure 1: Graph of $\operatorname{Sc}:\mathcal{M}^G_1\rightarrow{\Bbb R}$ in the variables $(x_1,x_2)$ for, from left to right, $M^{48}=\mathrm{SU}(5)\times\mathrm{SO}(8)/T^4$, $M^{21}=\mathrm{G}_2\times\mathrm{Sp}(2)/\mathrm{SU}(2)$ and $M^{29}=\mathrm{SU}(5)\times\mathrm{SU}(4)/\mathrm{Sp}(2)$, which admit one, two and none invariant Einstein metrics (i.e., critical points, in blue), respectively. The standard metric $g_{\operatorname{B}}$ ($x_1=x_2=1$) is in yellow and belongs to both the green curve of normal metrics and to the red curve defined by $x_1=x_2$.
  • Figure 2: Graph of the quartic polynomial $p$ whose roots are in bijection with invariant Einstein metrics on $M^{21}=\mathrm{G}_2\times\mathrm{Sp}(2)/\mathrm{SU}(2)$ (left) and $M^{29}=\mathrm{SU}(5)\times\mathrm{SU}(4)/\mathrm{Sp}(2)$ (right), which admit two and none, respectively.

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 28 more