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Summary: On isometric Lagrangian immersions
John Douglas Moore and Jean-Marie Morvan
Abstract
This article uses Cartan-Kšahler 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 three-manifolds which admit no local isomet-
ric Lagrangian immersions into complex three-space. It is expected that
isometric Lagrangian immersions of higher-dimensional Riemannian man-
ifolds 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
2
n(n + 1) functions of a single variable.
1 Introduction
This note is concerned with the question of which n-dimensional Riemannian
manifolds can be immersed isometrically as Lagrangian submanifolds of Cn
.
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