On the geometry of strong $G_2$-structures with torsion

by Mr Udhav Fowdar (Université de Turin)

Friday Jun 20, 2025, 11:15 AM → 12:15 PM Europe/Paris

112 (ICJ)

A strong geometry with torsion corresponds to a Riemannian manifold carrying a metric connection with closed skew-symmetric torsion. When this connection has reduced holonomy group $H$, then we say that the underlying $H$-structure is strong.
This notion of strong geometry with torsion has been predominantly studied in the context of Hermitian geometry, i.e. when $H=U(n)$; such manifolds are known as strong Kahler with torsion (SKT) or pluriclosed manifolds. In this talk, I will discuss the corresponding notion in the context of $G_2$ geometry. I will explain the analogy with (almost) SKT manifolds and give some new results characterising Ricci-flat strong $G_2$ manifolds with torsion. I will also explain how the same ideas can be applied to $6$-manifolds with suitable $SU(3)$-structures. In the spirit of making analogies with Hermitian geometry, I will also discuss a $G_2$ version of Gauduchon connections and the pluriclosed flow. This is based on joint works with Anna Fino.