Nash Manifolds
A Nash manifold denotes a real manifold furnished with algebraic structure, following a theorem of Nash that a compact differentiable manifold can be imbedded in a Euclidean space so that the image is precisely such a manifold. This book, in which almost
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1269
Masahiro Shiota
Nash Manifolds
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Masahiro Shiota Department of Mathematics, College of General Education Nagoya University, Nagoya, Japan
Mathematics Subject Classification (1980): 14G30, 58A07
ISBN 3-540-18102-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18102-4 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The purpose of these Lecture Notes is to construct a theory of real manifolds equipped with "algebraic" structures. Real nonsingular algebraic varieties display apparently singular phenomena, and this suggests we should not expect a unified theory to emerge from them. As we shall see, however, the objects we consider in these Notes form a broader class than the real nonsingularalgebraic varieties. A function on En is algebraic if and only if it is analytic and W
if the graph is semialgebraic in
Enx]R. We call such a function a C Na.6h 6unc-tion. We also call a Cr map with semialgebraic graph, O:$.r:$. w, a Cr Na.6h map. Then we define a Cr Na.6h mani60ld by sticking finite semialgebraic open sets in En together along semialgebraic c r Nash diffeomorphisms. It is the category of c r
open subsets by
Nash manifolds and Cr Nash maps that we are interested in.
The compact case was already studied well. In [N] Nash showed n l that a compact C manifold M can be imbedded in a Euclidean space m W
so that the image is a C
Nash submanifold of ]Rn.
He proved also that
W
Nash manifold structure on M is unique up to CW Nash diffeomorphism. Hence we can endow a compact Cl manifold with "algebraic" such a C
properties, which appears to contribute to differential topology. Indeed there are two applications [AM] and [K].
Hoping for more
applications, Palais developed a theory of compact affine CW Nash manifolds in [Pa].
We extend the object from CW Nash manifolds to Cr Nash manifolds. r This is because there always exists a partition of unity of c Nash W r class if 0 r < 00, a theorem of approximation of a C Nash map by a C Nash map holds true, and because the similarity and the contrast between the C interesting. and [SY].
o
Nash
and the C
W
Nash category are clear and
Our theory is based mainly on results in [Sh
l]
/ •.. / [Sh
S]
CONTENTS
1
INTRODUCTION Chapter I.
PRELIMINARIES
§I.l.
Semialgebraic sets
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