[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-45476-en":3,"doc-seo-45476-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},45476,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","A First Course in Geometric Topology and Differential Geometry","A comprehensive textbook guiding readers through geometric topology and differential geometry, starting with topological subsets of Euclidean space and core notions such as continuous maps, homeomorphisms, connectedness, and compactness. It develops topological and simplicial surfaces, including classification results, Euler characteristic, and simplicial curvature with the Gauss–Bonnet framework. Later chapters treat curves, smooth surfaces, curvature, geodesics, and proofs of the Gauss–Bonnet theorem, supported by appendices, references, and indices.",")事中  \nA First  \nCourse inGeometricTopologyand  \nDifferentialGeometryEthan D.BlochBirkhäuser  \nDedicated to my parents,in appreciation for all they have taught me  \nEthan D.Bloch  \n# A First Course\n\nin Geometric Topologyand Differential Geometry  \nBirkhäuserBoston·Basel·Berlin  \nLibrary of Congress Cataloging In-Publication Data  \nBloch,Ethan,1956-  \np.  cm.  \nPrinted on acid-free paper1997 Birkhäuser Boston  \nISBN 0-8176-3840-7  \nISBN 3-7643-3840-7  \nA first course in geometric topology and differential geometry/Ethan Bloch.  \nIncludes bibliographical references (p.·  )and index.  \nISBN 0-8176-3840-7(h:alk.paper).--ISBN 3-7643-3840-7(H:alk.paper)  \n1.Topology.2.Geometry,Differential.I.Title.  \nQA611.ZB55  199695-15470516.3'63-dc20CIP  \nBirkhäuser  \nCopyright is not claimed for works of U.S.Government employees.  \nAll rights reserved.No part of this publication may be reproduced,stored in a retrievalsystem,or transmitted,in any form or by any means,electronic,mechanical,photocopy-ing,recording,or otherwise,without prior permission of the copyright owner.  \nPermission to photocopy for internal or personal use of specific clients is granted byBirkhäuser Boston for libraries and other users registered with the Copyright ClearanceCenter(CCC),provided that the base fee of $6.00 per copy,plus $0.20 per page is paiddirectly to CCC,222 Rosewood Drive,Danvers,MA01923,U.S.A.Special requestsshould be addressed directly to Birkhäuser Boston,675 Massachusetts Avenue,Cam-bridge,MA02139,U.SA.  \nTypeset from author's disk in AMS-TEXby TEXniques,Inc.,Boston,MAIlustrations by Carl Twarog,Greenville,NCPrinted and bound by Maple-Vail,York,PAPrinted in the U.S.A.987654321  \nContents  \nIntroduction  \nix  \nTo the Student  \nxi  \nChapter I.Topology of Subsets of Euclidean Space  \n工  \n1.1 Introduction  \n1  \n1.2 Open and Closed Subsets of Sets in R\"2  \n1.3 Continuous Maps13  \n1.4 Homeomorphisms and Quotient Maps21  \n1.5 Connectedness27  \n1.6 Compactness34  \nChapter II.Topological Surfaces  \n47  \n2.1 Introduction  \n47  \n2.2 Arcs,Disks and 1-spheres  \n49  \n2.3 Surfaces in R\"55  \n2.4 Surfaces Via Gluing59  \n2.5 Properties of Surfaces  \n70  \n2.6 Connected Sum and the Classification of CompactConnected Surfaces73  \nAppendix A2.1 Proof of Theorem 2.4.3(i)82Appendix A2.2 Proof of Theorem 2.6.191  \nChapter III.Simplicial Surfaces  \n110  \n3.1 Introduction110  \n3.2 Simplices111  \n3.3 Simplicial Complexes119  \n3.4 Simplicial Surfaces131  \n3.5 The Euler Characteristic137  \n3.6 Proof of the Classification of Compact ConnectedSurfaces141  \n3.7 SimplicialCurvature and the Simplicial Gauss-BonnetTheorem152  \n## 3.8 Simplicial Disks and the Brouwer Fixed PointTheorem\n\n157  \n## Chapter IV.Curves in R³\n\n167  \n167  \n4.1 Introduction  \n4.2 Smooth Functions  \n167  \n4.3 Curves in R³  \n173  \n4.4 Tangent,Normal and Binormal Vectors  \n180  \n4.5 Curvature and Torsion  \n184  \n## 4.6 Fundamental Theorem of Curves\n\n192  \n4.7 Plane Curves  \n196  \n# Chapter V.Smooth Surfaces\n\n202  \n5.1 Introduction202  \n5.2 Smooth Surfaces202  \n5.3 Examples of Smooth Surfaces214  \n5.4 Tangent and Normal Vectors223  \n5.5 First Fundamental Form228  \n5.6 Directional Derivatives-Coordinate Free235  \n5.7 Directional Derivatives-Coordinates242  \n5.8 Length and Area252  \n5.9 Isometries257Appendix A5.1 Proof of Proposition 5.3.1229  \n270  \n# Chapter VI.Curvature of Smooth Surfaces\n\n270  \n6.1 Introduction and First Attempt  \n6.2 The Weingarten Map and the Second FundamentalForm274  \n6.3 Curvature-Second Attempt  \n281  \n6.4 Computations of Curvature Using Coordinates291  \n6.5 Theorema Egregium and the Fundamental Theoremof Surfaces296  \n# Chapter VII.Geodesics\n\n309  \n## 7.1 Introduction-\"Straight Lines\"on Surfaces\n\n309  \n7.2 Geodesics310  \n7.3 Shortest Paths322  \n## Chapter VIII.The Gauss-Bonnet Theorem\n\n328  \n8.1 Introduction328  \n8.2 The Exponential Map329  \n8.3 Geodesic Polar Coordinates335  \n8.4 Proof of the Gauss-Bonnet Theorem345  \n8.5 Non-Euclidean Geometry353Appendix A8.1 Geodesic Convexity362Appendix A8","cbCaihKLkeWzdaRy","https://ap.wps.com/l/cbCaihKLkeWzdaRy","pdf",9299109,6,1,440,"English","en",105,"# Introduction\n# To the Student\n# Chapter I. Topology of Subsets of Euclidean Space\n## Open and Closed Subsets of Sets in R^n\n## Continuous Maps\n## Homeomorphisms and Quotient Maps\n## Connectedness\n## Compactness\n# Chapter II. Topological Surfaces\n## Arcs, Disks and 1-spheres\n## Surfaces Via Gluing\n## Connected Sum and the Classification of Compact Connected Surfaces\n# Chapter III. Simplicial Surfaces\n## Simplicial Complexes\n## The Euler Characteristic\n## Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem\n# Chapter IV. Curves in R^3\n## Tangent, Normal and Binormal Vectors\n## Curvature and Torsion\n## Fundamental Theorem of Curves\n# Chapter V. Smooth Surfaces\n## First Fundamental Form\n## Length and Area\n## Isometries\n# Chapter VI. Curvature of Smooth Surfaces\n## The Weingarten Map and the Second Fundamental Form\n## Theorema Egregium and the Fundamental Theorem of Surfaces\n# Chapter VII. Geodesics\n## Geodesics\n## Shortest Paths\n# Chapter VIII. The Gauss-Bonnet Theorem\n## The Exponential Map\n## Proof of the Gauss-Bonnet Theorem\n## Non-Euclidean Geometry","[{\"question\":\"What topics are covered in the early chapters of the book?\",\"answer\":\"The book begins with topology of subsets of Euclidean space, including open/closed sets, continuous maps, homeomorphisms and quotient maps, connectedness, and compactness.\"},{\"question\":\"How does the book develop surfaces beyond basic topology?\",\"answer\":\"It introduces topological surfaces and then builds simplicial surfaces, covering simplices, simplicial complexes, the Euler characteristic, and simplicial curvature connected to the Gauss–Bonnet theorem.\"},{\"question\":\"Which later topics link geometry of surfaces to curvature and geodesics?\",\"answer\":\"Subsequent chapters treat curves in R^3, smooth surfaces, curvature via the Weingarten map and second fundamental form, and then geodesics and shortest paths, culminating in the Gauss–Bonnet theorem proof.\"}]",1783460055,1109,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-first-course-in-geometric-topology-and-differential-geometry","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-first-course-in-geometric-topology-and-differential-geometry/45476/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-19","2026-07-07",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What topics are covered in the early chapters of the book?","Question",{"text":76,"@type":77},"The book begins with topology of subsets of Euclidean space, including open/closed sets, continuous maps, homeomorphisms and quotient maps, connectedness, and compactness.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the book develop surfaces beyond basic topology?",{"text":81,"@type":77},"It introduces topological surfaces and then builds simplicial surfaces, covering simplices, simplicial complexes, the Euler characteristic, and simplicial curvature connected to the Gauss–Bonnet theorem.",{"name":83,"@type":74,"acceptedAnswer":84},"Which later topics link geometry of surfaces to curvature and geodesics?",{"text":85,"@type":77},"Subsequent chapters treat curves in R^3, smooth surfaces, curvature via the Weingarten map and second fundamental form, and then geodesics and shortest paths, culminating in the Gauss–Bonnet theorem 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