3 jul 2015 -- 15:00
Bielefeld University
Abstract.
The Hitchin flow equations in seven dimensions are certain 1st order pdes whose solutions define 8d Riemannian manifolds with holonomy contained in the exceptional holonomy group $Spin(7)$. Now the irreducible holonomy groups $Sp(2)$ and $SU(4)$ are contained in $Spin(7)$ and may so, in principal, also be obtained by the Hitchin flow.
In fact, it is known that the holonomy group is contained in $SU(4)$ if the Hitchin flow is induced by the hypo flow in seven dimensions.
We study both the Hitchin and the hypo flow in the left-invariant setting on a 7d Lie group and present some conditions on the initial values of these flows which ensure that the obtained Riemannian manifold has holonomy equal to $SU(4)$. We also discuss the case of holonomy equal to $Sp(2)$ and exclude this holonomy for certain initial values.