Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

Details Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3? Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

2003-04-18

arxiv.org/abs/gr-qc/0304066v3

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Misprints in eqs.(9-11) corrected. Submitted to Classical and Quantum Gravity

Scientific paper

10.1088/0264-9381/20/22/L01

Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on $K3$, or on surfaces whose universal covering is $K3$.

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