Submanifolds of complex quadrics and Gauss maps

The complex quadric and complex hyperbolic quadric are simply connected K\'ahler-Einstein manifolds. They can be constructed as K\'ahler submanifolds of the complex projective space and the complex anti-de Sitter space respectively, where the ambient spaces are equipped with the metric induced by the Hopf-fibration. The structure of the quadrics can be described in a very similar way and their curvature tensors are exactly opposite. In this thesis, we use a structural approach to investigate Lagrangian submanifolds of these quadrics and we completely classify their totally geodesic surfaces. In addition, a detailed description of these quadrics is given in dimension two, where they are isometric to a product of two-spheres and hyperbolic planes respectively.On both quadrics we introduce a family of non-integrable almost product structures. When studying the Lagrangian submanifolds we can use this almost product structure to define angle functions that describe most of the geometry of the Lagrangian submanifolds.In addition, we give a Gauss map of hypersurfaces of spheres, introduced by Palmer in \cite{Palmer1994}, which is a Lagrangian immersion in the complex quadric. Moreover, using this map the angle functions of the Lagrangian submanifold hold a relation to the principal curvatures of the hypersurface. Similarly, we define a Gauss map for spacelike hypersurfaces of anti-de Sitter space and show that it is a Lagrangian immersion in the complex hyperbolic quadric. These Gauss maps then allow us to relate the classification results of the Lagrangian submanifolds to these hypersurfaces.Using this Gauss map in \cite{Li2020} classifications are given of the minimal Lagrangian submanifolds of the complex quadric that have constant sectional curvature, with all $n$ angle functions constant and $n-1$ angle functions constant. We also discuss a new result for the Lagrangian submanifolds which are H-umbilical. For the complex hyperbolic quadric, we restrict ourselves to studying the minimal Lagrangian submanifolds with constant sectional curvature and those with all angle functions constant. There exist only six types of totally geodesic surfaces in the complex quadric and the complex hyperbolic quadric, we give an explicit description of all the possible types. This result has been shown by Chen, Nagano and Klein, but in this thesis we also present an alternative proof to show that these are the only possibilities. This approach uses the structures introduced throughout the thesis.

Jaar van publicatie:2021

Toegankelijkheid:Open