Submanifolds of four-dimensional Thurston geometries

In Riemannian geometry, submanifolds are natural objects to study. One often looks at submanifolds with special properties, for example ones that are totally geodesic, parallel, minimal etc. The existence or non-existence of certain submanifolds may give away information about the ambient space in which they live. Ambient spaces in which the study of submanifolds is of interest are Thurston geometries. These are homogeneous Riemannian manifolds that are of major importance in geometry and topology. The three-dimensional Thurston geometries appear as so-called building blocks for 3-manifolds in the Thurston geometrization conjecture. The proof of this conjecture was a big leap forward in mathematics in general, as it implies the three-dimensional Poincaré conjecture. This is the first Millennium Prize Problem to be solved. Given the importance of the three-dimensional geometries, we are led to study their four- dimensional counterparts in an attempt to tackle 4-manifolds with lower-dimensional techniques. In this research project, we aim to increase the understanding of four-dimensional Thurston geometries from a submanifold point of view.