Lagrangian Klein bottles in R^{2n}

TL;DR

This paper proves that n-dimensional Klein bottles can be Lagrangian embedded into R^{2n} only when n is odd, revealing a fundamental topological constraint for such embeddings.

Contribution

It establishes a necessary and sufficient condition for Lagrangian embeddings of Klein bottles in even-dimensional Euclidean spaces based on the parity of n.

Findings

Lagrangian Klein bottles exist in R^{2n} if and only if n is odd.

The result links topological properties of Klein bottles with symplectic embedding constraints.

Provides a complete characterization of Lagrangian embeddings for Klein bottles in Euclidean spaces.

Abstract

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd.