Equiaffine Characterization Of Lagrangian Surfaces in $R^4$

TL;DR

This paper characterizes Lagrangian surfaces in R^4 that are compatible with some parallel symplectic form, extending the understanding of their geometric properties and the affine normal plane bundle.

Contribution

It provides a new characterization of Lagrangian surfaces in R^4 based on their relation to parallel symplectic forms and the affine normal plane bundle.

Findings

Lagrangian surfaces have a special relation to the affine normal plane bundle.

The paper identifies conditions under which surfaces are Lagrangian with respect to some parallel symplectic form.

It extends the geometric understanding of non-degenerate surfaces in R^4.

Abstract

For non-degenerate surfaces in $R^4$, a distinguished transversal bundle called affine normal plane bundle was proposed in [Nomizu-Vrancken]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in $R^4$.