Infinite families of homogeneous Bismut Ricci flat manifolds

TL;DR

The paper constructs infinite families of compact homogeneous spaces carrying invariant BRF (Bismut Ricci-flat) pairs by starting from compact symmetric spaces of inner type. It identifies M = (G×G)/K_diag, with $(G^{\sigma})^{o} \subseteq K \subseteq G^{\sigma}$, as admitting invariant BRF structures, and provides a G×G-equivariant minimal embedding into G×G when K = G^σ, pulling back the standard bi-invariant metric and harmonic 3-form. The resulting BRF on M is Ricci flat but not flat, with the torsion form not parallel, and M has finite fundamental group with $b_3(M)=\ell$ (the number of simple factors of G); semisimple G yields a finite cover by products of inner symmetric-pair factors. Altogether, the work extends Cartan-type embeddings to 4-symmetric spaces, yielding new, nontrivial BRF homogeneous manifolds that are fixed points of generalized Ricci flow and enriching the landscape of BRF geometries in generalized geometry.

Abstract

Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order $4$ and (up to coverings) can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.

Theorems & Definitions (21)

• Corollary
• Proposition 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Lemma 2.4
• Lemma 2.5
• proof