GTA Seminar | Speaker: Juan Manuel Lorenzo Naveiro (University of Oklahoma, USA)

Title: Totally geodesic submanifolds of nearly Kähler and G₂ -manifolds.

Abstract: An almost Hermitian manifold (N²ⁿ,J) is nearly Kähler if the covariant derivative ∇J is totally skew-symmetric. Given a simply connected and (strictly) nearly Kähler 6-manifold N ≠ S⁶, one can rescale the metric on N in such a way that its cone Ñ has special holonomy G₂. A theorem of Butruille asserts that the simply connected, homogeneous, and strictly nearly Kähler manifolds of dimension 6 are  S⁶ , the projective space ℂP³, the flag manifold F(ℂ³) and the almost product S³×S³. These spaces fall under the class of naturally reductive homogeneous spaces, whose geometry can be understood in purely Lie-algebraic terms.

The aim of this talk is to describe the classification of totally geodesic submanifolds of the aforementioned spaces, as well as their Riemannian cones. In order to do this, we introduce the necessary tools to work with naturally reductive homogeneous spaces and cones, and afterwards we will describe the examples that appear in each case.

This talk is based on a joint work with Alberto Rodríguez Vázquez (Université Libre de Bruxelles, Belgium)