## Abstract

This article uses Cartan-Kähler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kähler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which admit no local isometric Lagrangian immersions into complex three-space. It is expected that isometric Lagrangian immersions of higher-dimensional Riemannian manifolds will almost always be unique. However, there is a plentiful supply of flat Lagrangian submanifolds of any complex n-space; we show that locally these depend on 1/2n(n + 1) functions of a single variable.

Original language | English (US) |
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Pages (from-to) | 833-849 |

Number of pages | 17 |

Journal | Illinois Journal of Mathematics |

Volume | 45 |

Issue number | 3 |

State | Published - Sep 2001 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics