# Riemannian manifolds with special holonomy and special curvature. Simons Center program "G2 manifolds". Simons Center for Geometry and Physics . September 30, 2014

Invited Talk

While the local generality of metrics with special holonomy is now well-understood and the existence of complete or even compact examples is well-established, the construction of explicit examples remains a challenge. An approach that has been followed with great success is to look for examples with large symmetry groups, for this often reduces the problem to studying systems of ordinary differential equations. A different approach is to look for examples whose curvature tensors satisfy some algebraically natural conditions, for this imposes higher order (partial) differential equations on the solutions, and so the methods of exterior differential systems can be brought to bear to analyze the resulting overdetermined systems of equations. This can sometimes yield solutions with relatively high cohomogeneity and may be expected to provide interesting characterizations of some of the known solutions as well. In this talk, I will report on work in progress along these lines, particularly in the low dimensions, in which the holonomy can be either $SU(2)$, $SU(3)$, or $G_2$. I will also show how these methods can be used to study similar problems in certain geometries with torsion.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- September 30, 2014

### Service or Event Name

- Simons Center program "G2 manifolds"

### Host Organization

- Simons Center for Geometry and Physics

### Location or Venue

- Simons Center for Geometry and Physics, Stony Brook, NY