differential topology - dundas

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1、DierentialTopologyBjørnIanDundas26thJune20022Contents1Preface72Introduction92.1Arobot'sarm:..................................92.1.1Question.................................112.1.2Dependenceonthetelescope'slength.................122.1.3Moral..................................132.2Furtherexamples.....

2、...........................142.2.1Charts..................................142.2.3Compactsurfaces............................152.2.8Higherdimensions...........................193Smoothmanifolds213.1Topologicalmanifolds..............................213.2Smoothstructures...............................2

3、43.3Maximalatlases.................................283.4Smoothmaps..................................303.5Submanifolds..................................353.6Productsandsums...............................394Thetangentspace434.0.1Predenitionofthetangentspace...................434.1Germs...............

4、.......................444.2Thetangentspace...............................474.3Derivations...................................525Vectorbundles575.1Topologicalvectorbundles...........................585.2Transitionfunctions...............................625.3Smoothvectorbundles........................

5、.....635.4Pre-vectorbundles...............................665.5Thetangentbundle...............................6834CONTENTS6Submanifolds756.1Therank.....................................756.2Theinversefunctiontheorem.........................776.3Theranktheorem................................786.4Regul

6、arvalues..................................816.5Immersionsandimbeddings..........................886.6Sard'stheorem.................................917Partitionofunity937.1Denitions....................................937.2Smoothbumpfunctions............................947.3Renementsofcoverings.....

7、.......................967.4Existenceofsmoothpartitionsofunityonsmoothmanifolds.........987.5ImbeddingsinEuclideanspace.........................998Constructionsonvectorbundles1018.1Subbundlesandrestri