New Solutions of the Einstein-Dirac Equation in Dimension n=3

TL;DR

This paper introduces a new family of metrics on the 3-sphere that admit WK-spinors, expanding the known solutions to the Einstein-Dirac equation in three dimensions and characterizing certain manifolds with constant Ricci eigenvalues.

Contribution

It identifies a one-parameter family of left-invariant metrics on S^3 with WK-spinors, including non-Einstein Sasakian metrics, and classifies related manifolds with constant Ricci eigenvalues.

Findings

Existence of a one-parameter family of metrics on S^3 with WK-spinors.

Includes non-Einstein Sasakian metrics with WK-spinors.

Characterization of certain 3-manifolds with constant Ricci eigenvalues.

Abstract

The aim of this short note is to announce the existence of a one-parameter family of left-invariant metrics on $S^3$ admitting WK-spinors. This family contains the two non-Einstein Sasakian metrics with WK-spinors on $S^3$, but does not contain the standard sphere $S^3$ with Killing spinors. Moreover, any simply-connected, complete Riemannian manifold $X^3 \not= S^3$ with WK-spinors such that the eigenvalues of the Ricci tensor are constant is isometric to a space of this one-parameter family.