Einstein metrics on \(S^ 3\), \(R^ 3\) and \(R^ 4\) bundles
Let M be a quaternionic Kähler manifold, i.e. a Riemannian manifold with holonomy group \(Hol\subset Sp(I)\cdot Sp(n),\) and \[ Z=\{J\in Sp(I)\quad \subset \quad GL(_{x}),\quad j^ 2=-id,\quad x\in M\} \] its twistor space. The authors consider the natural Riemannian metric \(g_{\lambda}\) on Z, depending on a real parameter \(\lambda\) and look for an Einstein metric in the form \(g=\alpha^ 2dr^ 2+\beta^ 2g_{\lambda},\) where \(\alpha\), \(\beta\), \(\lambda\) are functions of a new coordinate r. Then the Einstein equation reduces to a system of ODE. A numerical analysis indicates the existence of solutions of this system that can be extended to Einstein metrics on \(S^ 3\) bundles over a compact manifold M. Some explicit solutions are also obtained and their asymptotic properties are examined. These solutions define Ricci-flat cohomogeneity one metrics with holonomy groups \(G_ 2\) and Spin(7) on vector bundles over \(M=R^ 4\) or \(S^ 4\). The dimension of the moduli space of Ricci-flat metrics near a metric with holonomy \(G_ 2\) is calculated. It is proved also that on a manifold with holonomy \(G_ 2\) there is covariantly-constant spinor field and that any Riemannian manifold with covariantly-constant spinor field is Ricci-flat.
scientific article; zbMATH DE number 4147030