Classification of spherical Lagrangian submanifolds in complex Euclidean spaces

TL;DR

This paper provides a complete classification of spherical Lagrangian submanifolds in complex Euclidean spaces and extends the classification to Lagrangian submanifolds in complex pseudo-Euclidean spaces with arbitrary complex index.

Contribution

It offers the first comprehensive classification of spherical Lagrangian submanifolds in complex Euclidean spaces and extends results to pseudo-Euclidean settings.

Findings

Complete classification of spherical Lagrangian submanifolds in complex Euclidean spaces

Two classification theorems for Lagrangian submanifolds in complex pseudo-Euclidean spaces

Extension of classification results to spaces with arbitrary complex index

Abstract

An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a K\"ahler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each tangent space of $M^n$ with the corresponding normal space. In this paper, we completely classify spherical Lagrangian submanifolds in complex Euclidean spaces. Furthermore, we also provide two corresponding classification theorems for Lagrangian submanifolds in the complex pseudo-Euclidean spaces with arbitrary complex index.