國立高雄大學圖資館 |

Geometry of differential forms /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Geometry of differential forms /Shigeyuki Morita ; translated by Teruko Nagase, Katsumi Nomizu.
Author:	Morita, Shigeyuki,
Published:	Providence, R.I. :American Mathematical Society,©2001.
Description:	xxiv, 321 pages :illustrations ;22 cm
Subject:	Differential forms.
Online resource:	Kostenfrei
ISBN:	0821810456

Geometry of differential forms /
Morita, Shigeyuki,1946-

Geometry of differential forms /Shigeyuki Morita ; translated by Teruko Nagase, Katsumi Nomizu. - Providence, R.I. :American Mathematical Society,©2001. - xxiv, 321 pages :illustrations ;22 cm - Translations of mathematical monographs,v. 2010065-9282 ;. - Translations of mathematical monographs ;v. 201..

Includes bibliographical references (pages 315-316) and index.

Outline and Goal of the Theoryxix --

ISBN: 0821810456

Standard No.: 9780821810453

LCCN: 2001022608

Nat. Bib. Agency Control No.: 006567577UkSubjects--Topical Terms:

190957
Differential forms.

LC Class. No.: QA381 / .M6713 2001

Dewey Class. No.: 515/.37

Geometry of differential forms /
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020 $a 9780821810453 $q (softcover ; $q alk. paper)
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100 1 $a Morita, Shigeyuki, $d 1946- $3 883412
240 1 0 $a Bibun keishiki no kikagaku. $l English
245 1 0 $a Geometry of differential forms / $c Shigeyuki Morita ; translated by Teruko Nagase, Katsumi Nomizu.
260 $a Providence, R.I. : $b American Mathematical Society, $c ©2001.
300 $a xxiv, 321 pages : $b illustrations ; $c 22 cm
336 $a text $b txt $2 rdacontent
337 $a unmediated $b n $2 rdamedia
338 $a volume $b nc $2 rdacarrier
490 1 $a Translations of mathematical monographs, $x 0065-9282 ; $v v. 201
490 1 $a Iwanami series in modern mathematics
504 $a Includes bibliographical references (pages 315-316) and index.
505 0 0 $t Outline and Goal of the Theory $g xix -- $g Chapter 1 $t Manifolds $g 1 -- $g (a) $t N-dimensional numerical space R[superscript n] $g 2 -- $g (b) $t Topology of R[superscript n] $g 3 -- $g (c) $t C[infinity] functions and diffeomorphisms $g 4 -- $g (d) $t Tangent vectors and tangent spaces of R[superscript n] $g 6 -- $g (e) $t Necessity of an abstract definition $g 10 -- $g 1.2 $t Definition and examples of manifolds $g 11 -- $g (a) $t Local coordinates and topological manifolds $g 11 -- $g (b) $t Definition of differentiable manifolds $g 13 -- $g (c) $t R[superscript n] and general surfaces in it $g 16 -- $g (d) $t Submanifolds $g 19 -- $g (e) $t Projective spaces $g 21 -- $g (f) $t Lie groups $g 22 -- $g 1.3 $t Tangent vectors and tangent spaces $g 23 -- $g (a) $t C[infinity] functions and C[infinity] mappings on manifolds $g 23 -- $g (b) $t Practical construction of C[infinity] functions on a manifold $g 25 -- $g (c) $t Partition of unity $g 27 -- $g (d) $t Tangent vectors $g 29 -- $g (e) $t Differential of maps $g 33 -- $g (f) $t Immersions and embeddings $g 34 -- $g 1.4 $t Vector fields $g 36 -- $g (a) $t Vector fields $g 36 -- $g (b) $t Bracket of vector fields $g 38 -- $g (c) $t Integral curves of vector fields and one-parameter group of local transformations $g 39 -- $g (d) $t Transformations of vector fields by diffeomorphism $g 44 -- $g 1.5 $t Fundamental facts concerning manifolds $g 44 -- $g (a) $t Manifolds with boundary $g 44 -- $g (b) $t Orientation of a manifold $g 46 -- $g (c) $t Group actions $g 49 -- $g (d) $t Fundamental groups and covering manifolds $g 51 -- $g Chapter 2 $t Differential Forms $g 57 -- $g 2.1 $t Definition of differential forms $g 57 -- $g (a) $t Differential forms on R[superscript n] $g 57 -- $g (b) $t Differential forms on a general manifold $g 61 -- $g (c) $t Exterior algebra $g 61 -- $g (d) $t Various definitions of differential forms $g 66 -- $g 2.2 $t Various operations on differential forms $g 69 -- $g (a) $t Exterior product $g 69 -- $g (b) $t Exterior differentiation $g 70 -- $g (c) $t Pullback by a map $g 72 -- $g (d) $t Interior product and Lie derivative $g 72 -- $g (e) $t Cartan formula and properties of Lie derivatives $g 73 -- $g (f) $t Lie derivative and one-parameter group of local transformations $g 77 -- $g 2.3 $t Frobenius theorem $g 80 -- $g (a) $t Frobenius theorem--Representation by vector fields $g 80 -- $g (b) $t Commutative vector fields $g 82 -- $g (c) $t Proof of the Frobenius theorem $g 83 -- $g (d) $t Frobenius theorem--Representation by differential forms $g 86 -- $g (a) $t Differential forms with values in a vector space $g 89 -- $g (b) $t Maurer-Cartan form of a Lie group $g 90 -- $g Chapter 3 $t De Rham Theorem $g 95 -- $g 3.1 $t Homology of manifolds $g 96 -- $g (a) $t Homology of simplicial complexes $g 96 -- $g (b) $t Singular homology $g 99 -- $g (c) $t C[infinity] triangulation of C[infinity] manifolds $g 100 -- $g (d) $t C[infinity] singular chain complexes of C[infinity] manifolds $g 103 -- $g 3.2 $t Integral of differential forms and the Stokes theorem $g 104 -- $g (a) $t Integral of n-forms on n-dimensional manifolds $g 104 -- $g (b) $t Stokes theorem (in the case of manifolds) $g 107 -- $g (c) $t Integral of differential forms on chains, and the Stokes theorem $g 109 -- $g 3.3 $t De Rham theorem $g 111 -- $g (a) $t de Rham cohomology $g 111 -- $g (b) $t De Rham theorem $g 113 -- $g (c) $t Poincare lemma $g 116 -- $g 3.4 $t Proof of the de Rham theorem $g 119 -- $g (a) $t Cech cohomology $g 119 -- $g (b) $t Comparison of de Rham cohomology and Cech cohomology $g 121 -- $g (c) $t Proof of the de Rham theorem $g 126 -- $g (d) $t De Rham theorem and product structure $g 131 -- $g 3.5 $t Applications of the de Rham theorem $g 133 -- $g (a) $t Hopf invariant $g 133 -- $g (b) $t Massey product $g 136 -- $g (c) $t Cohomology of compact Lie groups $g 137 -- $g (d) $t Mapping degree $g 138 -- $g (e) $t Integral expression of the linking number by Gauss $g 140 -- $g Chapter 4 $t Laplacian and Harmonic Forms $g 145 -- $g 4.1 $t Differential forms on Riemannian manifolds $g 145 -- $g (a) $t Riemannian metric $g 145 -- $g (b) $t Riemannian metric and differentieal forms $g 148 -- $g (c) $t *-operator of Hodge $g 150 -- $g 4.2 $t Laplacian and harmonic forms $g 153 -- $g 4.3 $t Hodge theorem $g 158 -- $g (a) $t Hodge theorem and the Hodge decomposition of differential forms $g 158 -- $g (b) $t Idea for the proof of the Hodoge decomposition $g 160 -- $g 4.4 $t Applications of the Hodge theorem $g 162 -- $g (a) $t Poincare duality theorem $g 162 -- $g (b) $t Manifolds and Euler number $g 164 -- $g (c) $t Intersection number $g 165 -- $g Chapter 5 $t Vector Bundles and Characteristic Classes $g 169 -- $g 5.1 $t Vector bundles $g 169 -- $g (a) $t Tangent bundle of a manifold $g 169 -- $g (b) $t Vector bundles $g 170 -- $g (c) $t Several constructions of vector bundles $g 173 -- $g 5.2 $t Geodesics and parallel translation of vectors $g 180 -- $g (a) $t Geodesics $g 180 -- $g (b) $t Covariant derivative $g 181 -- $g (c) $t Parallel displacement of vectors and curvature $g 183 -- $g 5.3 $t Connections in vector bundles and $g 185 -- $g (a) $t Connections $g 185 -- $g (b) $t Curvature $g 186 -- $g (c) $t Connection form and curvature form $g 188 -- $g (d) $t Transformation rules of the local expressions for a connection and its curvature $g 190 -- $g (e) $t Differential forms with values in a vector bundle $g 191 -- $g 5.4 $t Pontrjagin classes $g 193 -- $g (a) $t Invariant polynomials $g 193 -- $g (b) $t Definition of Pontrjagin classes $g 197 -- $g (c) $t Levi-Civita connection $g 201 -- $g 5.5 $t Chern classes $g 204 -- $g (a) $t Connection and curvature in a complex vector bundle $g 204 -- $g (b) $t Definition of Chern classes $g 205 -- $g (c) $t Whitney formula $g 207 -- $g (d) $t Relations between Pontrjagin and Chern classes $g 208 -- $g 5.6 $t Euler classes $g 211 -- $g (a) $t Orientation of vector bundles $g 211 -- $g (b) $t Definition of the Euler class $g 211 -- $g (c) $t Properties of the Euler class $g 214 -- $g 5.7 $t Applications of characteristic classes $g 216 -- $g (a) $t Gauss-Bonnet theorem $g 216 -- $g (b) $t Characteristic classes of the complex projective space $g 223 -- $g (c) $t Characteristic numbers $g 225 -- $g Chapter 6 $t Fiber Bundles and Characteristic Classes $g 231 -- $g 6.1 $t Fiber bundle and principal bundle $g 231 -- $g (a) $t Fiber bundle $g 231 -- $g (b) $t Structure group $g 233 -- $g (c) $t Principal bundle $g 236 -- $g (d) $t Classification of fiber bundles and characteristic classes $g 238 -- $g (e) $t Examples of fiber bundles $g 239 -- $g 6.2 $t S[superscript 1] bundles and Euler classes $g 240 -- $g (a) $t S[superscript 1] bundle $g 241 -- $g (b) $t Euler class of an S[superscript 1] bundle $g 241 -- $g (c) $t Classification of S[superscript 1] bundles $g 246 -- $g (d) $t Defining the Euler class for an S[superscript 1] bundle by using differential forms $g 249 -- $g (e) $t Primary obstruction class and the Euler class of the sphere bundle $g 254 -- $g (f) $t Vector fields on a manifold and Hopf index theorem $g 255 -- $g 6.3 $t Connections $g 257 -- $g (a) $t Connections in general fiber bundles $g 257 -- $g (b) $t Connections in principal bundles $g 260 -- $g (c) $t Differential form representation of a connection in a principal bundle $g 262 -- $g 6.4 $t Curvature $g 265 -- $g (a) $t Curvature form $g 265 -- $g (b) $t Weil algebra $g 268 -- $g (c) $t Exterior differentiation of the Weil algebra $g 270 -- $g 6.5 $t Characteristic classes $g 275 -- $g (a) $t Weil homomorphism $g 275 -- $g (b) $t Invariant polynomials for Lie groups $g 279 -- $g (c) $t Connections for vector bundles and principal bundles $g 282 -- $g (d) $t Characterisric classes $g 284 -- $g 6.6 $t A couple of items $g 285 -- $g (a) $t Triviality of the cohomology of the Weil algebra $g 285 -- $g (b) $t Chern-Simons forms $g 287 -- $g (c) $t Flat bundles and holonomy homomorphisms $g 287.
650 0 $a Differential forms. $3 190957
650 0 $a Differentiable manifolds. $3 190948
650 6 $a Formes différentiables. $3 883451
650 6 $a Variétés différentiables. $2 ram $3 883452
830 0 $a Translations of mathematical monographs ; $v v. 201. $x 0065-9282 $3 883413
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