Homogeneous spin Riemannian manifolds with the simplest Dirac operator

TL;DR

This paper identifies a class of nonsymmetric homogeneous spin Riemannian manifolds with Dirac operators similar to symmetric spaces, characterized by traceless cyclic conditions and constructed via transversally symmetric fibrations.

Contribution

It characterizes and classifies nonsymmetric homogeneous spin Riemannian manifolds with symmetric-like Dirac operators using cyclic conditions and fibrations.

Findings

Existence of nonsymmetric homogeneous spin manifolds with symmetric Dirac operators.

Characterization of these manifolds as traceless cyclic with respect to certain quotients.

Provision of a classification list via transversally symmetric fibrations.

Abstract

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them.