A brief introduction to G2-geometry

Ponente:  Andrés Julián Moreno Ospina (UNICAMP)
Fecha:  martes 09 de abril de 2024 - 11:00-13:00
Lugar:  Aula Gris 2, ICMAT


G2-geometry arises as a special geometry on seven dimensional Riemannian manifolds. Roughly speaking, it consists in the existence of a smooth cross product on the tangent bundle of M, which is called the G2-structure. A remarkable feature is that the G2-structure encodes the whole Riemannian geometry of M, i.e. any Riemannian invariant depends on the G2-structure. And consequently, under some "technical hypothesis", this geometry appears as one of the cases of Berger's classification of the possible  Riemannian holonomy group of a metric.

This presentation is going to be divided into two sessions. In the first instance, we provide the linear algebra and the representation theory associated with the G2-geometry of a 7-manifold M, aiming to describe the first order G2-invariants of a Riemannian manifold and how these are related to the Riemannian invariants of M. As well as the relation of G2-geometry with other geometries, such as SU(3) and Spin(7) geometry.

In the second part, we review the construction of the first complete non-compact example of a metric with holonomy G2 given by Bryant and Salamon '89. And finally, we are going to mention some of the mathematical problems that arise in the context of G2-geometry, such as (if time allows it) G2-instantons, associative submanifolds and geometric flows.

Main references:

R. Bryant, Some remarks on G2-structures, Proceedings of 12th Gokova Geometry-Topology Conference (2005)
D. Joyce, Compact manifolds with special holonomy, Oxford University Press (2000)
S Karigiannis, Introduction to G2-geometry, Lectures and surveys on G2-manifolds and related topics, (2020)