Left-invariant Einstein metrics on S3 x S3
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=S3 x S3 together with the simplifying assumption that the isotropy group K of g in the group of motions is non-trivial.
When K≠Z2 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=Z2 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:
- 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").