Holdings: An introduction to manifolds

An introduction to manifolds:

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Bibliographic Details
Main Author:	Tu, Loring W. 1952- (Author)
Format:	Book
Language:	English
Published:	New York, NY Springer 2008
Series:	Universitext
Subjects:	Manifolds (Mathematics) Differentialform Mannigfaltigkeit Tangentialraum Vektorfeld Differenzierbare Mannigfaltigkeit Einführung Lehrbuch
Online Access:	Inhaltsverzeichnis
Physical Description:	XV, 360 S. graph. Darst.
ISBN:	9780387480985 0387480986 9780387481012

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adam_text	Contents Preface ......................................................... vii 0 A Brief Introduction .......................................... 1 Parti Euclidean Spaces 1 Smooth Functions on a Euclidean Space ......................... 5 1.1 C°° Versus Analytic Functions ............................... 5 1.2 Taylor s Theorem with Remainder ............................ 7 Problems ...................................................... 9 2 Tangent Vectors in R as Derivations ............................ 11 2.1 The Directional Derivative ................................... 12 2.2 Germs of Functions ......................................... 13 2.3 Derivations at a Point ....................................... 14 2.4 Vector Fields .............................................. 15 2.5 Vector Fields as Derivations .................................. 17 Problems ...................................................... 18 3 Alternating ^-Linear Functions ................................. 19 3.1 Dual Space ................................................ 19 3.2 Permutations .............................................. 20 3.3 Multilinear Functions ....................................... 22 3.4 Permutation Action on £ -Linear Functions ...................... 23 3.5 The Symmetrizing and Alternating Operators ................... 24 3.6 The Tensor Product ......................................... 25 3.7 The Wedge Product ......................................... 25 3.8 Anticommutativity of the Wedge Product ....................... 27 3.9 Associativity of the Wedge Product ............................ 28 3.10 ABasis for fc-Covectors ..................................... 30 Problems ...................................................... 31 χ Contents 4 Differential Forms on M ...................................... 33 4.1 Differential 1-Forms and the Differential of a Function ........... 33 4.2 Differential fc-Forms ........................................ 35 4.3 Differential Forms as Multilinear Functions on Vector Fields ...... 36 4.4 The Exterior Derivative ..................................... 36 4.5 Closed Forms and Exact Forms ............................... 39 4.6 Applications to Vector Calculus ............................... 39 4.7 Convention on Subscripts and Superscripts ..................... 42 Problems ...................................................... 42 Partii Manifolds 5 Manifolds ................................................... 47 5.1 Topological Manifolds ...................................... 47 5.2 Compatible Charts .......................................... 48 5.3 Smooth Manifolds .......................................... 50 5.4 Examples of Smooth Manifolds .............................. 51 Problems ...................................................... 53 6 Smooth Maps on a Manifold ................................... 57 6.1 Smooth Functions and Maps ................................. 57 6.2 Partial Derivatives .......................................... 60 6.3 The Inverse Function Theorem ............................... 60 Problems ...................................................... 62 7 Quotients ................................................... 63 7.1 The Quotient Topology ...................................... 63 7.2 Continuity of a Map on a Quotient ............................ 64 7.3 Identification of a Subset to a Point ............................ 65 7.4 A Necessary Condition for a Hausdorff Quotient ................ 65 7.5 Open Equivalence Relations ................................. 66 7.6 The Real Projective Space ................................... 68 7.7 The Standard C°° Atlas on a Real Projective Space .............. 71 Problems ...................................................... 73 Partili The Tangent Space 8 The Tangent Space ........................................... 77 8.1 The Tangent Space at a Point ................................. 77 8.2 The Differential of a Map .................................... 78 8.3 The Chain Rule ............................................ 79 8.4 Bases for the Tangent Space at a Point ......................... 80 8.5 Local Expression for the Differential .......................... 82 8.6 Curves in a Manifold ....................................... 83 Contents xi 8.7 Computing the Differential Using Curves ...................... 85 8.8 Rank, Critical and Regular Points ............................. 86 Problems ...................................................... 87 9 Submanifolds ................................................ 91 9.1 Submanifolds .............................................. 91 9.2 The Zero Set of a Function ................................... 94 9.3 The Regular Level Set Theorem .............................. 95 9.4 Examples of Regular Submanifolds ........................... 97 Problems ...................................................... 98 10 Categories and Functors ...................................... 101 10.1 Categories ................................................ 101 10.2 Functors .................................................. 102 10.3 Dual Maps ................................................ 103 Problems ...................................................... 104 11 The Rank of a Smooth Map .................................... 105 11.1 Constant Rank Theorem ..................................... 106 11.2 Immersions and Submersions ................................ 107 11.3 Images of Smooth Maps ..................................... 109 11.4 Smooth Maps into a Submanifold ............................. 113 11.5 The Tangent Plane to a Surface in M3 .......................... 115 Problems ...................................................... 116 12 The Tangent Bundle .......................................... 119 12.1 The Topology ofthe Tangent Bundle .......................... 119 12.2 The Manifold Structure on the Tangent Bundle .................. 121 12.3 Vector Bundles ............................................. 121 12.4 Smooth Sections ........................................... 123 12.5 Smooth Frames ............................................ 125 Problems ...................................................... 126 13 Bump Functions and Partitions of Unity ......................... 127 13.1 C°° Bump Functions ........................................ 127 13.2 Partitions of Unity .......................................... 131 13.3 Existence of aPartition of Unity .............................. 132 Problems ...................................................... 134 14 Vector Fields ................................................ 135 14.1 Smoothness of a Vector Field ................................. 135 14.2 Integral Curves ............................................ 136 14.3 Local Flows ............................................... 138 14.4 The Lie Bracket ............................................ 141 14.5 Related Vector Fields ....................................... 143 14.6 The Push-Forward of a Vector Field ........................... 144 Problems ...................................................... 144 Contents Part IV Lie Groups and Lie Algebras 15 Lie Groups .................................................. 149 15.1 Examples of Lie Groups ..................................... 149 15.2 Lie Subgroups ............................................. 152 15.3 The Matrix Exponential ..................................... 153 15.4 The Trace of a Matrix....................................... 155 15.5 The Differential of det at the Identity .......................... 157 Problems ...................................................... 157 16 Lie Algebras ................................................. 161 16.1 Tangent Space at the Identity of a Lie Group .................... 161 16.2 The Tangent Space to SL(n, R) at I ........................... 161 16.3 The Tangent Space to O(n) at ƒ .............................. 162 16.4 Left-Invariant Vector Fields on a Lie Group .................... 163 16.5 The Lie Algebra of a Lie Group ............................... 165 16.6 The Lie Bracket on gl(n, Ж) .................................. 166 16.7 The Push-Forward of a Left-Invariant Vector Field ............... 167 16.8 The Differential as a Lie Algebra Homomorphism ............... 168 Problems ...................................................... 170 PartV Differential Forms 17 Differential 1-Forms .......................................... 175 17.1 The Differential of a Function ................................ 175 17.2 Local Expression for a Differential 1-Form ..................... 176 17.3 The Cotangent Bundle ...................................... 177 17.4 Characterization of C°° 1-Forms .............................. 177 17.5 Pullback of 1-forms ......................................... 179 Problems ...................................................... 179 18 Differential ¿-Forms .......................................... 181 18.1 Local Expression for a ł-Form ............................... 182 18.2 The Bundle Point of View ................................... 183 18.3 C^/fc-Forms .............................................. 183 18.4 Pullback of ¿-Forms ........................................ 184 18.5 The Wedge Product ......................................... 184 18.6 Invariant Forms on a Lie Group ............................... 186 Problems ...................................................... 186 Contents xiii 19 The Exterior Derivative .......................................189 19.1 Exterior Derivative on a Coordinate Chart ...................... 190 19.2 Local Operators ............................................ 190 19.3 Extension of a Local Form to a Global Form .................... 191 19.4 Existence of an Exterior Differentiation ........................ 192 19.5 Uniqueness of Exterior Differentiation ......................... 192 19.6 The Restriction of a fc-Form to a Submanifold ................... 193 19.7 A Nowhere-Vanishing 1-Form on the Circle .................... 193 19.8 Exterior Differentiation Under a Pullback ...................... 195 Problems ...................................................... 196 Part VI Integration 20 Orientations .................................................201 20.1 Orientations on a Vector Space ............................... 201 20.2 Orientations and и -Covectors ................................ 203 20.3 Orientations on a Manifold ................................... 204 20.4 Orientations and Atlases ..................................... 206 Problems ...................................................... 208 21 Manifolds with Boundary .....................................211 21.1 Invariance of Domain ....................................... 211 21.2 Manifolds with Boundary .................................... 213 21.3 The Boundary of a Manifold with Boundary .................... 215 21.4 Tangent Vectors, Differential Forms, and Orientations ............ 215 21.5 Boundary Orientation for Manifolds of Dimension Greater than One .................................................. 216 21.6 Boundary Orientation for One-Dimensional Manifolds ........... 218 Problems ...................................................... 219 22 Integration on a Manifold .....................................221 22.1 The Riemann Integral of a Function on K ..................... 221 22.2 Integrability Conditions ..................................... 223 22.3 The Integral of an η -Form on W .............................. 224 22.4 The Integral of a Differential Form on a Manifold ............... 225 22.5 Stokes Theorem ........................................... 228 22.6 Line Integrals and Green s Theorem ........................... 230 Problems ...................................................... 231 Part VII De Rham Theory 23 De Rham Cohomology ........................................235 23.1 De Rham Cohomology ...................................... 235 23.2 Examples of de Rham Cohomology ........................... 237 23.3 Diffeomorphism Invariance .................................. 239 23.4 The Ring Structure on de Rham Cohomology ................... 240 Problems ...................................................... 242 xiv Contents 24 The Long Exact Sequence in Cohomology ........................243 24.1 Exact Sequences ........................................... 243 24.2 Cohomology of Cochain Complexes .......................... 245 24.3 The Connecting Homomorphism .............................. 246 24.4 The Long Exact Sequence in Cohomology ..................... 247 Problems ...................................................... 248 25 The Mayer-Vietoris Sequence ..................................249 25.1 The Mayer-Vietoris Sequence ................................ 249 25.2 The Cohomology of the Circle ................................ 253 25.3 The Euler Characteristic ..................................... 254 Problems ...................................................... 255 26 Homotopy Invariance .........................................257 26.1 Smooth Homotopy ......................................... 257 26.2 Homotopy Type ............................................ 258 26.3 Deformation Retractions ..................................... 260 26.4 The Homotopy Axiom for de Rham Cohomology ................ 261 Problems ...................................................... 262 27 Computation of de Rham Cohomology ..........................263 27.1 Cohomology Vector Space of a Torus .......................... 263 27.2 The Cohomology Ring of a Torus ............................. 265 27.3 The Cohomology of a Surface of Genus g ...................... 267 Problems ...................................................... 271 28 Proof of Homotopy Invariance .................................273 28.1 Reduction to Two Sections ................................... 274 28.2 Cochain Homotopies ........................................ 274 28.3 Differential Forms on Μ χ Ж................................ 275 28.4 A Cochain Homotopy Between ΐζ and ij1 ....................... 276 28.5 Verification of Cochain Homotopy ............................ 276 Part VIII Appendices A Point-Set Topology ........................................... 281 A. 1 Topological Spaces ......................................... 281 A.2 Subspace Topology ......................................... 283 A.3 Bases ..................................................... 284 A.4 Second Countability ........................................ 285 A.5 Separation Axioms ......................................... 286 A.6 The Product Topology ....................................... 287 A.7 Continuity ................................................ 289 A.8 Compactness .............................................. 290 Contents xv A.9 Connectedness ............................................. 293 АЛО Connected Components ..................................... 294 АЛ 1 Closure ................................................... 295 АЛ2 Convergence .............................................. 296 Problems ...................................................... 297 В The Inverse Function Theorem on Ш and Related Results ..........299 ВЛ The Inverse Function Theorem ............................... 299 B.2 The Implicit Function Theorem ............................... 300 B.3 Constant Rank Theorem ..................................... 303 Problems ...................................................... 304 С Existence of a Partition of Unity in General ......................307 D Linear Algebra ..............................................311 D. 1 Linear Transformations ...................................... 311 D.2 Quotient Vector Spaces ...................................... 312 Solutions to Selected Exercises Within the Text .......................315 Hints and Solutions to Selected End-of-Chapter Problems ..............319 List of Symbols ..................................................339 References ......................................................347 Index ...........................................................349
adam_txt	Contents Preface . vii 0 A Brief Introduction . 1 Parti Euclidean Spaces 1 Smooth Functions on a Euclidean Space . 5 1.1 C°° Versus Analytic Functions . 5 1.2 Taylor's Theorem with Remainder . 7 Problems . 9 2 Tangent Vectors in R" as Derivations . 11 2.1 The Directional Derivative . 12 2.2 Germs of Functions . 13 2.3 Derivations at a Point . 14 2.4 Vector Fields . 15 2.5 Vector Fields as Derivations . 17 Problems . 18 3 Alternating ^-Linear Functions . 19 3.1 Dual Space . 19 3.2 Permutations . 20 3.3 Multilinear Functions . 22 3.4 Permutation Action on £ -Linear Functions . 23 3.5 The Symmetrizing and Alternating Operators . 24 3.6 The Tensor Product . 25 3.7 The Wedge Product . 25 3.8 Anticommutativity of the Wedge Product . 27 3.9 Associativity of the Wedge Product . 28 3.10 ABasis for fc-Covectors . 30 Problems . 31 χ Contents 4 Differential Forms on M" . 33 4.1 Differential 1-Forms and the Differential of a Function . 33 4.2 Differential fc-Forms . 35 4.3 Differential Forms as Multilinear Functions on Vector Fields . 36 4.4 The Exterior Derivative . 36 4.5 Closed Forms and Exact Forms . 39 4.6 Applications to Vector Calculus . 39 4.7 Convention on Subscripts and Superscripts . 42 Problems . 42 Partii Manifolds 5 Manifolds . 47 5.1 Topological Manifolds . 47 5.2 Compatible Charts . 48 5.3 Smooth Manifolds . 50 5.4 Examples of Smooth Manifolds . 51 Problems . 53 6 Smooth Maps on a Manifold . 57 6.1 Smooth Functions and Maps . 57 6.2 Partial Derivatives . 60 6.3 The Inverse Function Theorem . 60 Problems . 62 7 Quotients . 63 7.1 The Quotient Topology . 63 7.2 Continuity of a Map on a Quotient . 64 7.3 Identification of a Subset to a Point . 65 7.4 A Necessary Condition for a Hausdorff Quotient . 65 7.5 Open Equivalence Relations . 66 7.6 The Real Projective Space . 68 7.7 The Standard C°° Atlas on a Real Projective Space . 71 Problems . 73 Partili The Tangent Space 8 The Tangent Space . 77 8.1 The Tangent Space at a Point . 77 8.2 The Differential of a Map . 78 8.3 The Chain Rule . 79 8.4 Bases for the Tangent Space at a Point . 80 8.5 Local Expression for the Differential . 82 8.6 Curves in a Manifold . 83 Contents xi 8.7 Computing the Differential Using Curves . 85 8.8 Rank, Critical and Regular Points . 86 Problems . 87 9 Submanifolds . 91 9.1 Submanifolds . 91 9.2 The Zero Set of a Function . 94 9.3 The Regular Level Set Theorem . 95 9.4 Examples of Regular Submanifolds . 97 Problems . 98 10 Categories and Functors . 101 10.1 Categories . 101 10.2 Functors . 102 10.3 Dual Maps . 103 Problems . 104 11 The Rank of a Smooth Map . 105 11.1 Constant Rank Theorem . 106 11.2 Immersions and Submersions . 107 11.3 Images of Smooth Maps . 109 11.4 Smooth Maps into a Submanifold . 113 11.5 The Tangent Plane to a Surface in M3 . 115 Problems . 116 12 The Tangent Bundle . 119 12.1 The Topology ofthe Tangent Bundle . 119 12.2 The Manifold Structure on the Tangent Bundle . 121 12.3 Vector Bundles . 121 12.4 Smooth Sections . 123 12.5 Smooth Frames . 125 Problems . 126 13 Bump Functions and Partitions of Unity . 127 13.1 C°° Bump Functions . 127 13.2 Partitions of Unity . 131 13.3 Existence of aPartition of Unity . 132 Problems . 134 14 Vector Fields . 135 14.1 Smoothness of a Vector Field . 135 14.2 Integral Curves . 136 14.3 Local Flows . 138 14.4 The Lie Bracket . 141 14.5 Related Vector Fields . 143 14.6 The Push-Forward of a Vector Field . 144 Problems . 144 Contents Part IV Lie Groups and Lie Algebras 15 Lie Groups . 149 15.1 Examples of Lie Groups . 149 15.2 Lie Subgroups . 152 15.3 The Matrix Exponential . 153 15.4 The Trace of a Matrix. 155 15.5 The Differential of det at the Identity . 157 Problems . 157 16 Lie Algebras . 161 16.1 Tangent Space at the Identity of a Lie Group . 161 16.2 The Tangent Space to SL(n, R) at I . 161 16.3 The Tangent Space to O(n) at ƒ . 162 16.4 Left-Invariant Vector Fields on a Lie Group . 163 16.5 The Lie Algebra of a Lie Group . 165 16.6 The Lie Bracket on gl(n, Ж) . 166 16.7 The Push-Forward of a Left-Invariant Vector Field . 167 16.8 The Differential as a Lie Algebra Homomorphism . 168 Problems . 170 PartV Differential Forms 17 Differential 1-Forms . 175 17.1 The Differential of a Function . 175 17.2 Local Expression for a Differential 1-Form . 176 17.3 The Cotangent Bundle . 177 17.4 Characterization of C°° 1-Forms . 177 17.5 Pullback of 1-forms . 179 Problems . 179 18 Differential ¿-Forms . 181 18.1 Local Expression for a ł-Form . 182 18.2 The Bundle Point of View . 183 18.3 C^/fc-Forms . 183 18.4 Pullback of ¿-Forms . 184 18.5 The Wedge Product . 184 18.6 Invariant Forms on a Lie Group . 186 Problems . 186 Contents xiii 19 The Exterior Derivative .189 19.1 Exterior Derivative on a Coordinate Chart . 190 19.2 Local Operators . 190 19.3 Extension of a Local Form to a Global Form . 191 19.4 Existence of an Exterior Differentiation . 192 19.5 Uniqueness of Exterior Differentiation . 192 19.6 The Restriction of a fc-Form to a Submanifold . 193 19.7 A Nowhere-Vanishing 1-Form on the Circle . 193 19.8 Exterior Differentiation Under a Pullback . 195 Problems . 196 Part VI Integration 20 Orientations .201 20.1 Orientations on a Vector Space . 201 20.2 Orientations and и -Covectors . 203 20.3 Orientations on a Manifold . 204 20.4 Orientations and Atlases . 206 Problems . 208 21 Manifolds with Boundary .211 21.1 Invariance of Domain . 211 21.2 Manifolds with Boundary . 213 21.3 The Boundary of a Manifold with Boundary . 215 21.4 Tangent Vectors, Differential Forms, and Orientations . 215 21.5 Boundary Orientation for Manifolds of Dimension Greater than One . 216 21.6 Boundary Orientation for One-Dimensional Manifolds . 218 Problems . 219 22 Integration on a Manifold .221 22.1 The Riemann Integral of a Function on K" . 221 22.2 Integrability Conditions . 223 22.3 The Integral of an η -Form on W . 224 22.4 The Integral of a Differential Form on a Manifold . 225 22.5 Stokes' Theorem . 228 22.6 Line Integrals and Green's Theorem . 230 Problems . 231 Part VII De Rham Theory 23 De Rham Cohomology .235 23.1 De Rham Cohomology . 235 23.2 Examples of de Rham Cohomology . 237 23.3 Diffeomorphism Invariance . 239 23.4 The Ring Structure on de Rham Cohomology . 240 Problems . 242 xiv Contents 24 The Long Exact Sequence in Cohomology .243 24.1 Exact Sequences . 243 24.2 Cohomology of Cochain Complexes . 245 24.3 The Connecting Homomorphism . 246 24.4 The Long Exact Sequence in Cohomology . 247 Problems . 248 25 The Mayer-Vietoris Sequence .249 25.1 The Mayer-Vietoris Sequence . 249 25.2 The Cohomology of the Circle . 253 25.3 The Euler Characteristic . 254 Problems . 255 26 Homotopy Invariance .257 26.1 Smooth Homotopy . 257 26.2 Homotopy Type . 258 26.3 Deformation Retractions . 260 26.4 The Homotopy Axiom for de Rham Cohomology . 261 Problems . 262 27 Computation of de Rham Cohomology .263 27.1 Cohomology Vector Space of a Torus . 263 27.2 The Cohomology Ring of a Torus . 265 27.3 The Cohomology of a Surface of Genus g . 267 Problems . 271 28 Proof of Homotopy Invariance .273 28.1 Reduction to Two Sections . 274 28.2 Cochain Homotopies . 274 28.3 Differential Forms on Μ χ Ж. 275 28.4 A Cochain Homotopy Between ΐζ and ij1 . 276 28.5 Verification of Cochain Homotopy . 276 Part VIII Appendices A Point-Set Topology . 281 A. 1 Topological Spaces . 281 A.2 Subspace Topology . 283 A.3 Bases . 284 A.4 Second Countability . 285 A.5 Separation Axioms . 286 A.6 The Product Topology . 287 A.7 Continuity . 289 A.8 Compactness . 290 Contents xv A.9 Connectedness . 293 АЛО Connected Components . 294 АЛ 1 Closure . 295 АЛ2 Convergence . 296 Problems . 297 В The Inverse Function Theorem on Ш" and Related Results .299 ВЛ The Inverse Function Theorem . 299 B.2 The Implicit Function Theorem . 300 B.3 Constant Rank Theorem . 303 Problems . 304 С Existence of a Partition of Unity in General .307 D Linear Algebra .311 D. 1 Linear Transformations . 311 D.2 Quotient Vector Spaces . 312 Solutions to Selected Exercises Within the Text .315 Hints and Solutions to Selected End-of-Chapter Problems .319 List of Symbols .339 References .347 Index .349
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genre	(DE-588)4151278-9 Einführung gnd-content (DE-588)4123623-3 Lehrbuch gnd-content
genre_facet	Einführung Lehrbuch
id	DE-604.BV023012373
illustrated	Illustrated
index_date	2024-07-02T19:09:59Z
indexdate	2024-07-09T21:08:57Z
institution	BVB
isbn	9780387480985 0387480986 9780387481012
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016216574
oclc_num	186358733
open_access_boolean
owner	DE-703 DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-20 DE-11
owner_facet	DE-703 DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-20 DE-11
physical	XV, 360 S. graph. Darst.
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Springer
record_format	marc
series2	Universitext
spelling	Tu, Loring W. 1952- Verfasser (DE-588)110090322 aut An introduction to manifolds Loring W. Tu New York, NY Springer 2008 XV, 360 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Manifolds (Mathematics) Differentialform (DE-588)4149772-7 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Tangentialraum (DE-588)4792364-7 gnd rswk-swf Vektorfeld (DE-588)4139571-2 gnd rswk-swf Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content (DE-588)4123623-3 Lehrbuch gnd-content Differentialform (DE-588)4149772-7 s Tangentialraum (DE-588)4792364-7 s Vektorfeld (DE-588)4139571-2 s DE-604 Mannigfaltigkeit (DE-588)4037379-4 s Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016216574&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Tu, Loring W. 1952- An introduction to manifolds Manifolds (Mathematics) Differentialform (DE-588)4149772-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Tangentialraum (DE-588)4792364-7 gnd Vektorfeld (DE-588)4139571-2 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd
subject_GND	(DE-588)4149772-7 (DE-588)4037379-4 (DE-588)4792364-7 (DE-588)4139571-2 (DE-588)4012269-4 (DE-588)4151278-9 (DE-588)4123623-3
title	An introduction to manifolds
title_auth	An introduction to manifolds
title_exact_search	An introduction to manifolds
title_exact_search_txtP	An introduction to manifolds
title_full	An introduction to manifolds Loring W. Tu
title_fullStr	An introduction to manifolds Loring W. Tu
title_full_unstemmed	An introduction to manifolds Loring W. Tu
title_short	An introduction to manifolds
title_sort	an introduction to manifolds
topic	Manifolds (Mathematics) Differentialform (DE-588)4149772-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Tangentialraum (DE-588)4792364-7 gnd Vektorfeld (DE-588)4139571-2 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd
topic_facet	Manifolds (Mathematics) Differentialform Mannigfaltigkeit Tangentialraum Vektorfeld Differenzierbare Mannigfaltigkeit Einführung Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016216574&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT tuloringw anintroductiontomanifolds

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