# Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres

## TL;DR

This paper explores the relationship between hypersurfaces in spheres and Lagrangian submanifolds in the complex quadric, providing explicit constructions and revealing a connection between principal curvatures and angle functions.

## Contribution

It offers explicit methods to construct hypersurfaces from Lagrangian submanifolds and establishes a novel relation between their geometric properties.

## Key findings

- Explicit constructions of the Gauss map correspondence.
- A relation between principal curvatures and angle functions.
- Multiple hypersurfaces correspond to the same Lagrangian submanifold.

## Abstract

The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian submanifold of $Q^n$ is locally the image under the Gauss map of several hypersurfaces of $S^{n+1}(1)$. In this paper, we give explicit constructions for these correspondences and we prove a relation between the principal curvatures of a hypersurface of $S^{n+1}(1)$ and the local angle functions of the corresponding Lagrangian submanifold of $Q^n$. The existence of such a relation is remarkable since the definition of the angle functions depends on the choice of an almost product structure on $Q^n$ and since several hypersurfaces of $S^{n+1}(1)$, with different principal curvatures, correspond to the same Lagrangian submanifold of $Q^n$.