 
Summary: On isometric Lagrangian immersions
John Douglas Moore and JeanMarie Morvan
Abstract
This article uses CartanKšahler theory to show that a small neighbor
hood 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 Kšahler surface). In fact, such immersions depend on
two functions of a single variable. On the other hand, explicit examples
are given of Riemannian threemanifolds which admit no local isomet
ric Lagrangian immersions into complex threespace. It is expected that
isometric Lagrangian immersions of higherdimensional Riemannian man
ifolds will almost always be unique. However, there is a plentiful supply
of flat Lagrangian submanifolds of any complex nspace; we show that
locally these depend on 1
2
n(n + 1) functions of a single variable.
1 Introduction
This note is concerned with the question of which ndimensional Riemannian
manifolds can be immersed isometrically as Lagrangian submanifolds of Cn
.
