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Ricci curvature and Einstein metrics on aligned homogeneous spaces

Jorge Lauret, Cynthia Will

Abstract

Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so called {\it aligned}, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of $G$-invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces.

Ricci curvature and Einstein metrics on aligned homogeneous spaces

Abstract

Let be a compact homogeneous space and assume that and have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that if has simple factors, so called {\it aligned}, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of -invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces.

Paper Structure

This paper contains 18 sections, 19 theorems, 206 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Aligned homogeneous spaces
  3. Definition
  4. Large classes
  5. Reductive decompositions
  6. Normal metrics
  7. Ricci and scalar curvature
  8. Normal Einstein metrics
  9. The case $s=2$
  10. Diagonal metrics
  11. The case of $M=G_1\times G_2/K$
  12. Ricci curvature
  13. Standard diagonalization
  14. Einstein metrics
  15. The case $\mathcal{M}^{diag}=\mathcal{M}^G$
  16. ...and 3 more sections

Key Result

Theorem 1.1

A normal metric on an aligned space $M=G/K$ is never Einstein, unless $M$ is the Ledger-Obata space $K\times\dots\times K/\Delta K$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 58 more