The Metric Theory of Banach Manifolds
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662
Ethan Akin
The Metric Theory of Banach Manifolds
Springer-Verlag Berlin Heidelberg New York 1978
Author Ethan Akin Mathematics Department The City College 137 Street and Convent Avenue New York City, N.Y. 10031/USA
Library of Congress Cataloging in Publication Data
Akin, Ethan, 1946The metric theory of Banach manifolds. (Lecture notes in mathematics ; 662) Bibliography: p. Includes index. 1. Differentiable manifolds. 2. Mappings (Mathematics) 3. Geometry, Differential. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 662. QA3.L28 no. 662 [QA614.3) 510'.8s [516'.36) 78-14728
AMS Subject Classifications (1970): 58A05, 58B20, 53C60, 58015, 57030 ISBN 3-540-08915-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08915-2 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
To:
Joyce Liz Sam
TABLE OF CONTENTS
Introduction
v
Chapter I: Atlases Sec. 1. Banach Spaces and Jets Sec. 2. Atlas Constructions
1
Chapter II: Function Spaces Elementary properties Sec. L Sec. 2. Composition Results Sec. 3. Local Homeomorphisms and Differential Equations Chapter III: Sec. L Sec. 2. Sec. 3. Sec. 4. Sec. 5. Sec. 6. Sec. 7. Sec. 8.
Metric Structures Atlases and Maps Metrics Finslers Metrics and Bounds Metric Structures Regular pseudometric spaces Semi completeness Exponential Maps
5 14 29 37 44 49 55 61 64 77 86 100
Chapter IV: Manifolds of Mappinqs Sec. L Section Spaces Sec. 2. Composition Results Sec. 3. Manifolds of Mappings Sec. 4. Linear Map Bundles Sec. 5. Tangent Bundle Identifications Sec. 6. Composition and Evaluation Results
107 114 120 134 142 146
Chapter V: Uniform and Halder Continuity Sec. 1. Accessory Pseudometric Structures Sec. 2. Admissible APM's on Semicomplete Spaces Sec. 3. Section Spaces Sec. 4. II Mappings
154 162 170 172
Chapter VI: Immersions and Submersions Sec. L Linear Injections and Surjections Sec. 2. VB Injections and Surjections Sec. 3. Immersions and Embeddings Sec. 4. Submersions Sec. 5. Local Isomorphisms Sec. 6. Transversality
174 182 190 203 207 211
Chapter VII: Sec. L Sec. 2. Sec. 3. Sec. 4.
Grassmannians and Leaf Immersions Grassmannian of a B-Space Grassmannian of a Bundle Smoothing Immersions Leaf Immersions
227 239 243 248
VI
Chapter VIII: Sec. Sec. Sec. Sec. Sec.
1. 2. 3. 4. 5.
Bibliography Index
Finite Dimensional Manifolds Manifolds of Finite Type Approximation and Smoothing and Finslers Semicompact Structures Compact Sets and Compact Mani
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