Left-invariant Einstein metrics on S^{3} x S^{3}

Florin Belgun, Vicente Cortés, Alexander S. Haupt, David Lindemann

Florin Belgun, Vicente Cortés, Alexander S. Haupt, David Lindemann

In arXiv:1703.10512 we continued the classification of homogeneous compact Einstein manifolds in dimension six.
We considered the remaining open case, namely left-invariant Einstein metrics g on G=S^{3} x S^{3} together with the simplifying assumption that the isotropy group K of g in the group of motions is non-trivial.
When K≠Z_{2} we prove that the Einstein metrics on G are given by (up to homothety) either the standard metric or the nearly Kähler metric, based on representation-theoretic arguments and computer algebra.
For the remaining case K=Z_{2} we present partial results.

The corresponding data, which consists of the input and output files of the computer-based Gröbner basis computations, is contained in the following files:

The corresponding data, which consists of the input and output files of the computer-based Gröbner basis computations, is contained in the following files:

- Gröbner basis input/output in Mathematica/Magma format used to solve eqs. (3.7), (3.13), and eq. (3.18) for fixed values of μ
as a compressed ZIP-file
(3.7MB compressed, 18.8MB uncompressed).

- Grevlex Gröbner basis data in Magma format used to examine eq. (3.18).

Input as a compressed ZIP-file (844 bytes compressed, 1.4KB uncompressed).

Output as a collection of 10 split compressed ZIP-files each 4GB in size,

file 01, file 02, file 03, file 04, file 05, file 06, file 07, file 08, file 09, file 10

(Total: 38.6GB compressed, 105.3GB uncompressed. Note: before unzipping, the split files need to be merged into a single file e.g. by using the command "cat 1703.10512gbdata2output.zip.* > 1703.10512gbdata2output.zip").