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1.

図書

図書
edited by Anthony Bak, Masaharu Morimoto, and Fumihiro Ushitaki
出版情報: Dordrecht : Kluwer Academic Publishers, c2002  xi, 249 p. ; 25 cm
シリーズ名: K-monographs in mathematics ; v. 7
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Introduction
Hilbert's fifth problem and proper actions of Lie groups / S. Illman1.:
Equivariant algebraic vector bundles over representations--a survey / M. Masuda2.:
G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories / T. Kawakami3.:
Geometry of Finite Topological spaces and equivariant finite topological spaces / S. Kono ; F. Ushitaki4.:
On the theory of homotopy representations. A survey / I. Nagasaki5.:
Manifolds as fixed point sets of smooth compact Lie group actions / K. Pawalowski6.:
Surgery and homotopy theory in study of the transformation groups / Y. Kitada7.:
Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres / 8.:
The Burnside ring revisited / M. Morimoto9.:
Multiplicative stabilization and transformation groups / S. Kwasik ; R. Schultz10.:
Symmetries on manifolds, deformations and rational homotopy: a survey / M. Raussen11.:
Rigidity of codimension one locally free actions of solvable Lie groups / N. Tsuchiya ; A. Yamakawa12.:
Smooth actions of non-compact semi-simple Lie groups / F. Uchida ; K. Mukoyama13.:
Hamiltonian group actions and equivariant indices / T. Takakura14.:
Controlled methods in equivariant topology, a survey / E. Pedersen15.:
Index
Introduction
Hilbert's fifth problem and proper actions of Lie groups / S. Illman1.:
Equivariant algebraic vector bundles over representations--a survey / M. Masuda2.:
2.

図書

図書
Guy David
出版情報: Basel : Birkhäuser, c2005  xiv, 581 p. ; 24 cm
シリーズ名: Progress in mathematics ; v. 233
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Foreword
Presentation of the Mumford-Shah functional
Functions in the Sobolev spaces
Regularity properties for quasiminimizers
Limits of almost-minimizers
Pieces of C^1 curves for almost-minimizers
Global Mumford-Shah minimizers in the plane
Applications to almost-minimizers (n = 2)
Quasi- and almost-minimizers in higher dimensions
Boundary regularity
Foreword
Presentation of the Mumford-Shah functional
Functions in the Sobolev spaces
3.

図書

図書
C.T.C. Wall
出版情報: Providence, RI : American Mathematical Society, c1999  xv, 302 p. ; 26 cm
シリーズ名: Mathematical surveys and monographs ; v. 69
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Preliminaries: Note on conventions
Basic homotopy notions Surgery below the middle dimension
Appendix: Applications Simple Poincare complexes
The main theorem: Statement of results
An important special case
The even-dimensional case
The odd-dimensional case
The bounded odd-dimensional case
The bounded even-dimensional case
Completion of the proof Patterns of application: Manifold structures on Poincare complexes
Applications to submanifolds Submanifolds: Other techniques
Separating submanifolds Two-sided submanifolds One-sided submanifolds
Calculations and applications: Calculations: Surgery obstruction groups
Calculations: The surgery obstructions
Applications: Free actions on spheres
General remarks An extension of the Atiyah-Singer $G$-signature theorem
Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces
Applications: Free uniform actions on euclidean space
Fake tori Polycyclic groups
Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work
function space methods Topological manifolds
Poincare embeddings Homotopy and simple homotopy
Further calculations Sullivan's results
Reformulations of the algebra Rational surgery
References
Index
Preliminaries: Note on conventions
Basic homotopy notions Surgery below the middle dimension
Appendix: Applications Simple Poincare complexes
4.

図書

図書
Charles Li, Stephen Wiggins
出版情報: New York : Springer, 1997  viii, 170 p. ; 25 cm
シリーズ名: Applied mathematical sciences ; v. 128
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5.

図書

図書
Bert-Wolfgang Schulze, Boris Sternin, Victor Shatalov
出版情報: Berlin ; Chichester : Wiley-VCH, c1998  376 p. ; 25 cm
シリーズ名: Mathematical topics ; v. 15
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6.

図書

図書
by Anatoliy K. Prykarpatsky and Ihor V. Mykytiuk
出版情報: Dordrecht : Kluwer Academic Press, c1998  553 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 443
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Preface
Background Notations
Dynamical systems with homogeneous configuration spaces / 1:
Dynamical systems with symmetries
Poisson structures / 1.1:
Lie group actions on manifolds / 1.2:
Symplectic structures on coadjoint representation orbits / 1.3:
Hamiltonian group actions / 1.4:
Invariant Hamiltonian systems with homogeneous configuration spaces / 1.5:
The existence of a maximal involutive set of functions on the orbits of semi-simple elements of a semi-simple Lie algebra / 2:
The algebra of invariant polynomials on a semi-simple Lie algebra / 2.1:
Semi-simple element orbits / 2.2:
Maximal involutive sets of functions on semi-simple elements orbits / 2.3:
The integrability criterion and spherical pairs of Lie groups / 3:
Criterion of integrability / 3.1:
Spherical pairs of complex Lie groups / 3.2:
Interpolation property of spherical pairs of compact Lie groups / 4:
Properties of spherical pairs of compact Lie algebras / 4.1:
Points in general position / 4.2:
Spherical pairs of classical simple Lie groups / 5:
Preliminary remarks / 5.1:
Involutions of simple Lie algebras / 5.2:
Spherical pairs of classical Lie algebras / 5.3:
Classification of spherical pairs of the exceptional simple Lie algebras / 6:
Classification of spherical pairs of semi-simple Lie groups / 7:
Spherical pairs of semi-simple Lie algebras / 7.1:
Geometric quantization and integrable dynamical systems
Connections on line bundles
Line bundles
Equivalence classes of line bundles
Connections
The integrality condition
Hermitian structures
Equivalence classes of line bundles with connections / 1.6:
Holomorphic line bundles / 1.7:
Derivations / 1.8:
Tensor products, square roots and invariant Hermitian structures / 1.9:
Parallel transport on a line bundle with a connection / 1.10:
Parallel transport and derivations / 1.11:
Flat partial connections
Flat F-connections
Maps of line bundles
Line bundles of differential forms
Geometric quantization
Polarizations
Geometric quantization and reduction
Introduction
Hamiltonian reduction
Quantization / 4.3:
Examples: geometric quantization of the oscillator type Hamiltonian systems
Geometric quantization of the generalized n-dimensional harmonic oscillator
Geometric quantization of geodesic flow on a sphere
Geometric quantization of the multidimensional Kepler problem
Geometric quantization of the MIC-Kepler problem / 5.4:
Structures on manifolds and algebraic integrability of dynamical systems
Poisson structures and dynamical systems with symmetries
Preliminary notes
Poisson structures on manifolds
Casimir functions and involution
Casimir functions associated with classical expansions of semi-simple Lie algebras
Alder-Kostant-Symes and Mishchenko-Fomenko theorems
The reduction method and Poisson structures on dual spaces of semi-direct sums of Lie algebras
The mapping canonicity of symplectic structures
Momentum mapping
Poisson structures on dual spaces of semi-direct sums of Lie algebras
Canonical mappings / 2.4:
Nonlinear Neumann type dynamical systems as integrable flows on coadjoint orbits of Lie groups
The Neumann problem
The Lie-Poisson bracket associated with the ad-semidirect sum of Lie algebras
The canonical symplectic structure on T(S[superscript n-1]) and its diffeomorphisms / 3.3:
An involutive system of integrals for the Neumann dynamical system on the sphere S[superscript n-1] / 3.4:
Generalized Neumann-Bogoliubov dynamical systems / 3.5:
Abelian integrals, integrable dynamical systems, and their Lax type representations
The Neumann-Rosochatius-Bogoliubov Hamiltonian system
Conservation laws
Lax type representation
Dual momentum mappings and their applications
Preliminaries
Dualities
The Neumann-Rosochatius system
The Lie algebraic setting of Benney-Kaup dynamical systems and associated via Moser Neumann-Bogoliubov oscillatory flows
The Novikov-Lax finite-dimensional invariant reductions on nonlocal submanifolds / 6.1:
The Moser map and its associated dual moment maps into loop Lie algebras / 6.3:
The finite-dimensional Moser type of reduction of modified Boussinesq and super-Korteweg-de Vries Hamiltonian systems via the gradient-holonomic algorithm and dual moment maps
The Moser type of finite-dimensional reduction of a Boussinesq hydrodynamic system and its Lie-algebraic integrability / 7.2:
The Neumann type of oscillatory super-Hamiltonian systems on the sphere S[superscript N] and their Lie algebraic super-integrability / 7.3:
Lax-type of flows on Grassmann manifolds and dual momentum mappings / 8:
Symplectic structures on loop Grassmann manifolds / 8.1:
An intrinsic loop Grassmannian structure and dual momentum mappings / 8.3:
On the geometric structure of integrable flows in Grassmann manifolds / 9:
Centrally extended symplectic structures and integrable flows on the loop Grassmann manifolds / 9.1:
Algebraic methods of quantum statistical mechanics and their applications
Current algebra representation formalism in nonrelativistic quantum mechanics
The current algebra in nonrelativistic quantum mechanics
Current algebra representations
Bogoliubov-Araki generating functional
Lie current algebra, Hamiltonian operator, and Bogoliubov functional equations
Hamiltonian operator
Gibbs states and the Kubo-Martin-Schwinger conditions
Stable states and the KMS condition
Functional-operator representations of the current Lie algebra
The Bogoliubov-Bloch functional equation / 2.5:
The reconstruction via Araki of the Hamiltonian operator and the Bogoluibov functional equation / 2.6:
Functional-operator solutions of the Bogoliubov functional equations / 2.7:
A generalized Virasoro algebra / 2.8:
The secondary quantization method and the spectrum of quantum excitations of a nonlinear Schrodinger type dynamical system
Preliminary notions
The second quantization representation
A generalized nonlinear Schrodinger type quantum dynamical system
The quantum inverse spectral transform method
The scattering operator
Eigenvalue states of the nonlinear Schrodinger type model / 3.6:
Quantum excitations of a bose gas with a positive momentum / 3.7:
Unitary representations of the generalized Virasoro algebra
Verma modules over the generalized Virasoro current algebra
Unitary irreducible modules with highest weight
Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds
The current Lie algebra on S[superscript 1] and its functional representations
Basic notations
Associated cohomology complexes and their properties
Differential geometry analysis on real jet manifolds
Differential geometry analysis on jet supermanifolds
Euler variational derivative, external differential and tensors on infinite-dimensional functional spaces
Implectic operators and Poisson structures
Dynamical systems and bi-Hamiltonicity
The equivalence of dynamical systems as the Backlund transformation
The current Lie algebra on a cycle as a symmetry subalgebra of compatibly bi-Hamiltonian nonlinear dynamical systems on an axis
The Hamiltonicity of nonlinear dynamical systems on infinite-dimensional functional manifolds
A Lie-algebraic algorithm for investigating integrability
The gradient holonomic algorithm and Lax type representation
An adjoint Lax type equation and conservation laws
Lax type representation: differential algebraic approach
Lax type representation: geometric approach
Lagrangian and Hamiltonian formalisms for reduced infinite-dimensional dynamical systems with symmetries
General setting
Lagrangian reduction
Symplectic analysis and Hamiltonian fields
Discrete dynamical systems. One generalization
Non-isospectrally integrable dynamical systems: the generalized asymptotic structure of conservation laws
A nonstandard reduction problem
Lagrangian and Hamiltonian analysis of dynamical systems on functional manifolds. The Poisson-Dirac bracket / 3.8:
Conclusions / 3.9:
The algebraic structure of the gradient-holonomic algorithm for Lax type integrable nonlinear dynamical systems
Algebraic structure of the Lax type integrable dynamical system
The periodic problem and canonical variational relationships
The spectral gradient structure of Lax integrable many-dimensional nonlinear dynamical systems on operator manifolds / 4.4:
The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra
An effective Maurer-Cartan one-form construction
General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra
Cartan's invariant geometric object structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems in partial derivatives
A loop algebra and the Yang-Baxter structure / 5.5:
The transfer matrix properties / 5.6:
The algebraic structure of the gradient-holonomic algorithm for the Lax-type nonlinear dynamical systems: the reduction via Dirac and the canonical quantization procedure
The generalized R-structure hierarchy
The Dirac quantization procedure for Moser induced finite-dimensional Neumann type Hamiltonian systems
The scalar and operator integrable Hamiltonian systems via the algebraic gradient-holonomic algorithm / 6.4:
Concluding remarks / 6.5:
Hamiltonian structures of hydrodynamical Benny type dynamical systems and their associated Boltzmann-Vlasov kinetic equations on an axis
The Boltzmann equation and an associated moment problem
A nonlinear completely integrable Schrodinger type dynamical system approximation
The complete integrability of a Benney type hydrodynamical system associated with a Boltzmann-Vlasov equation / 7.4:
Conclusion / 7.5:
Appendix
Basic definitions, examples / .1:
The tangent Lie algebra / .2:
Lie subgroups / .3:
Lie algebras / .4:
Cartan subalgebras / .5:
Semi-simple complex Lie algebras / .6:
References
Preface
Background Notations
Dynamical systems with homogeneous configuration spaces / 1:
7.

図書

図書
Alejandro Adem, Johann Leida, Yongbin Ruan
出版情報: Cambridge : Cambridge University Press, 2007  xi, 149 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 171
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Introduction
Foundations / 1:
Classical effective orbifolds / 1.1:
Examples / 1.2:
Comparing orbifolds to manifolds / 1.3:
Groupoids / 1.4:
Orbifolds as singular spaces / 1.5:
Cohomology, bundles and morphisms / 2:
De Rham and singular cohomology of orbifolds / 2.1:
The orbifold fundamental group and covering spaces / 2.2:
Orbifold vector bundles and principal bundles / 2.3:
Orbifold morphisms / 2.4:
Classification of orbifold morphisms / 2.5:
Orbifold K-theory / 3:
Orbifolds, group actions, and Bredon cohomology / 3.1:
Orbifold bundles and equivariant K-theory / 3.3:
A decomposition for orbifold K-theory / 3.4:
Projective representations, twisted group algebras, and extensions / 3.5:
Twisted equivariant K-theory / 3.6:
Twisted orbifold K-theory and twisted Bredon cohomology / 3.7:
Chen-Ruan cohomology / 4:
Twisted sectors / 4.1:
Degree shifting and Poincare pairing / 4.2:
Cup product / 4.3:
Some elementary examples / 4.4:
Chen-Ruan cohomology twisted by a discrete torsion / 4.5:
Calculating Chen-Ruan cohomology / 5:
Abelian orbifolds / 5.1:
Symmetric products / 5.2:
References
Index
Introduction
Foundations / 1:
Classical effective orbifolds / 1.1:
8.

図書

図書
Igor Nikolaev
出版情報: Berlin ; Heiderlberg : Springer, c2001  xxvi, 450 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, v. 41
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Index of Notation
Foliations on 2-Manifolds / 0:
Notations / 0.1:
Examples / 0.2:
Smooth Functions / 0.2.1:
1-Forms / 0.2.2:
Line Elements / 0.2.3:
Curvature Lines / 0.2.4:
A-Diffeomorphisms / 0.2.5:
Constructions / 0.3:
Suspension / 0.3.1:
Measured Foliations / 0.3.2:
Affine Foliations / 0.3.3:
Labyrinths / 0.3.4:
Gluing Together / 0.3.5:
General Theory / Part I:
Local Theory / 1:
Introduction / 1.1:
Symmetry / 1.2:
Normal Forms / 1.3:
Typical Normal Forms / 1.3.1:
Degenerate Normal Forms / 1.3.2:
Structurally Stable Singularities / 1.4:
Blowing-up Method / 1.4.1:
Fundamental Lemma / 1.4.2:
Classification / 1.4.3:
Bifurcations / 1.5:
Morse-Smale Foliations / 2:
Rough Foliations / 2.1:
Main Theorem / 2.1.1:
Structural Stability / 2.1.2:
Density / 2.1.3:
Classification of Morse-Smale Foliations / 2.2:
Rotation Systems / 2.2.1:
Equivalence Criterion / 2.2.2:
Realization of the Graphs / 2.2.3:
Example / 2.2.4:
Gradient-like Foliations / 2.3:
Lyapunov Function / 2.3.1:
Lyapunov Graph / 2.3.2:
Connected Components of Morse-Smale Foliations / 2.4:
Degrees of Stability / 2.5:
Foliations Without Holonomy / 3:
Periodic Components / 3.1:
Quasiminimal Sets / 3.2:
Structure of a Quasiminimal Set / 3.2.1:
Blowing-Down / 3.2.2:
Decomposition / 3.3:
Surgery / 3.4:
Surgery of Labyrinths / 3.4.1:
Surgery of Measured Foliations / 3.4.2:
Number of Quasiminimal Sets / 3.5:
Application: Smoothing Theorem / 3.6:
Invariants of Foliations / 4:
Torus / 4.1:
Minimal Foliations / 4.1.1:
Foliations With a Cantor Minimal Set / 4.1.2:
Foliations With Cherry Cells / 4.1.3:
Analytic Classification / 4.1.4:
Homotopy Rotation Class / 4.2:
Surfaces of Genus g ≥ 2 / 4.2.1:
Properties of the Homotopy Rotation Class / 4.2.2:
Non Orientable Surfaces / 4.3:
Torus With the Cross-Cap / 4.3.1:
Surfaces of Genus p ≥ 4 / 4.3.2:
Discrete Invariants / 4.4:
Regular Foliations on the Sphere / 4.4.1:
Orbit Complex / 4.4.2:
Cells / 4.5:
Classification of Elementary Cells / 4.5.2:
Amalgamation of Elementary Cells / 4.5.3:
Conley-Lyapunov-Peixoto Graph / 4.5.4:
Foliations With Symmetry / 4.5.5:
Cayley Graph / 4.6.1:
Isomorphism / 4.6.2:
Realization / 4.6.3:
Homology and Cohomology Invariants / 4.7:
Asymptotic Cycles / 4.7.1:
Fundamental Class / 4.7.2:
Cycles of A. Zorich / 4.7.3:
Smooth Classification / 4.8:
Torus and Klein Bottle / 4.8.1:
Surfaces of Genus g ≥ 2 / 4.8.2:
Curves on Surfaces / 5:
Curves and the Absolute / 5.1:
Background / 5.1.1:
Proof of Weil's Conjectures / 5.1.3:
Theorems of D. V. Anosov / 5.1.4:
Asymptotic Directions / 5.2:
Of Recurrent Semi-Trajectory / 5.2.1:
Of Analytic Flow / 5.2.2:
Of Foliation / 5.2.3:
Of Curves With Restriction on the Geodesic Curvature / 5.2.4:
Approximation of a Curve / 5.3:
Limit Sets at the Absolute / 5.4:
Geodesic Deviation / 5.5:
Deviation Property of Trajectories / 5.5.1:
Deviation From the Geodesic Framework / 5.5.2:
Ramified Coverings / 5.5.3:
Swing of Trajectories / 5.5.4:
Unbounded Deviation / 5.6:
Irrational Direction on Torus / 5.6.1:
Rational Direction on Torus / 5.6.3:
Family of Curves / 5.7:
Non-compact Surfaces / 6:
Foliations in the Plane / 6.1:
Non Singular Case / 6.1.1:
Singular Case / 6.1.2:
Level Set of Harmonic Functions / 6.1.3:
Depth of the Centre / 6.2:
Minimal Sets / 6.3.2:
Minimal Flows / 6.3.3:
Transitive Flows / 6.3.4:
Applications / Part II:
Ergodic Theory / 7:
Existence of Invariant Measures / 7.1:
Liouville's Theorem / 7.2.1:
Ergodicity / 7.2.2:
Mixing / 7.3.1:
Entropy / 7.4.1:
Homeomorphisms of the Unit Circle / 8:
Denjoy Flow / 8.1:
Cherry Class / 8.2:
Cherry Example / 8.2.1:
Flows With One Cell / 8.2.2:
Flows With Several Cells / 8.2.3:
Foliations on the Sphere / 8.3:
Main Result / 8.3.1:
Application to the Labyrinths / 8.3.3:
Appendix: The Dulac Functions / 8.3.4:
Addendum: Bendixson's Theorem / 8.4:
Diffeomorphisms of Surfaces / 9:
A-diffeomorphisms / 9.1:
Attractors of R. V. Plykin / 9.1.1:
One-Dimensional Basic Sets on the Sphere / 9.1.2:
Surfaces of Genus g ≥ 1 / 9.1.3:
Singularity Data / 9.2:
Isotopy Classes of Diffeomorphisms / 9.3:
C*-Algebras / 10:
Irrational Rotation Algebra / 10.1:
Dimension Groups / 10.1.1:
Continued Fractions / 10.1.2:
Effros-Shen's Theorem / 10.1.3:
Projections of Aα / 10.1.4:
Morita Equivalence / 10.1.5:
Embedding of Aα / 10.1.6:
Artin Rotation Algebra / 10.2:
Approximationssatz / 10.2.1:
Artin Numbers / 10.2.2:
K Theory / 10.2.3:
Foliation With Reeb Components / 10.3.1:
Baum-Connes Conjecture / 10.3.2:
C*-Algebras of Morse-Smale Flows / 10.4:
Quadratic Differentials / 11:
Finite Critical Points / 11.1:
Pole of Order 2 / 11.2.3:
Higher Order Poles / 11.2.4:
Global Behaviour of the Trajectories / 11.3:
Flat Structures / 12:
Flat Metric With Cone Singularities / 12.1:
Classification of Closed Flat Surfaces / 12.1.1:
Connection With Quadratic Differentials and Measured Foliations / 12.2:
Rational Billiards / 12.3:
Veech Dichotomy / 12.4:
Principal Curvature Lines / 13:
Invariants of the 2-Jets / 13.1:
Stability Lemma / 13.1.3:
Classification of Simple Umbilics / 13.1.4:
Carathéodory Conjecture / 13.2:
ϕ-Geodesics / 13.2.1:
CMC-Surfaces / 13.2.3:
Proof of Theorem 13.2.1 / 13.2.4:
Elements of Global Theory / 13.3:
Bifurcation of Umbilical Connections / 13.3.1:
Differential Equations / 14:
Characteristic Curve / 14.1:
Background and Notations / 14.1.1:
Theorem of Hartman and Wintner / 14.1.2:
Generic Singularities / 14.2:
Theorem of A. G. Kuzmin / 14.2.2:
Positive Differential 2-Forms / 15:
Space of Forms / 15.1:
Stable Subspace / 15.3.1:
Theorem of V. Guinez / 15.3.2:
Control Theory / B. Piccoli16:
Optimal Control / 16.1:
Optimal Flows / 16.3:
Generic Optimal Flows on the Plane / 16.4:
Optimal Flows on 2 Manifolds / 16.5:
Appendix / Part III:
Riemann Surfaces / 17:
Uniformization Theorem / 17.1:
Discrete Groups / 17.2:
Möbius Transformations / 17.2.1:
Fuchsian Group / 17.2.2:
Limit Set of Fuchsian Groups / 17.2.3:
Modular Group / 17.2.4:
Teichmuller Space / 17.2.5:
Conformal Invariants / 17.3.1:
Quasiconformal Mappings / 17.3.2:
Beltrami Equation / 17.3.3:
Ahlfors-Bers' Theorem / 17.3.4:
Geometry of Quadratic Differentials / 17.3.5:
Associated Metric / 17.3.6:
Isothermal Coordinates / 17.3.7:
Complex Curves / 17.4:
Projective Curves / 17.4.1:
Degree-Genus Formula / 17.4.2:
Elliptic Curves / 17.4.3:
Divisors and the Riemann-Roch Theorem / 17.4.4:
Application: Dimension of the Teichmuller Space / 17.4.5:
Bibliography
Index
Index of Notation
Foliations on 2-Manifolds / 0:
Notations / 0.1:
9.

図書

図書
Dominic D. Joyce
出版情報: Oxford ; New York : Oxford University Press, 2000  xii, 436 p. ; 25 cm
シリーズ名: Oxford mathematical monographs
Oxford science publications
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10.

図書

図書
Anders Kock
出版情報: Cambridge : Cambridge University Press, 2010  xiii, 302 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 180
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Preface
Acknowledgements
Calculus and linear algebra / 1:
The number line R / 1.1:
The basic infinitesimal spaces / 1.2:
The KL axiom scheme / 1.3:
Calculus / 1.4:
Affine combinations of mutual neighbour points / 1.5:
Geometry of the neighbour relation / 2:
Manifolds / 2.1:
Framings and 1-forms / 2.2:
Affine connections / 2.3:
Affine connections from framings / 2.4:
Bundle connections / 2.5:
Geometric distributions / 2.6:
Jets and jet bundles / 2.7:
Infinitesimal simplicial and cubical complex of a manifold / 2.8:
Combinatorial differential forms / 3:
Simplicial, whisker, and cubical forms / 3.1:
Coboundary/exteripr derivative / 3.2:
Integration of forms / 3.3:
Uniqueness of observables / 3.4:
Wedge/cup product / 3.5:
Involutive-distnbutions and differential forms / 3.6:
Non-abelian theory of 1-forms / 3.7:
Differential forms with values in a vector bundle / 3.8:
Crossed modules and non-abelian 2-forms / 3.9:
The tangent bundle / 4:
Tangent vectors and vector fields / 4.1:
Addition of tangent vectors / 4.2:
The log-exp bijection / 4.3:
Tangent vectors as differential operators / 4.4:
Cotangents, and the cotangent bundle / 4.5:
The differential operator of a linear connection / 4.6:
Classical differential forms / 4.7:
Differential forms with values in TM?M / 4.8:
Lie bracket of vector fields / 4.9:
Further aspects of the tangent bundle / 4.10:
Groupoids / 5:
Connections in groupoids / 5.1:
Actions of groupoids on bundles / 5.3:
Lie derivative / 5.4:
Deplacements in groupoids / 5.5:
Principal bundles / 5.6:
Principal connections / 5.7:
Holonomy of connections / 5.8:
Lie theory; non-abelian covariant derivative / 6:
Associative algebras / 6.1:
Differential forms with values in groups / 6.2:
Differential forms with values in a group bundle / 6.3:
Bianchi identity in terms of covariant derivative / 6.4:
Semidirecl products; covariant derivative as curvature / 6.5:
The Lie algebra of G / 6.6:
Group-valued vs. Lie-algebra-valued forms / 6.7:
Left-invariant distributions / 6.8:
Examples of enveloping algebras and enveloping algebra bundles / 6.10:
Jets and differential operators / 7:
Linear differential operators and their symbols / 7.1:
Linear deplacements as differential operators / 7.2:
Bundle-theoretic differential operators / 7.3:
Sheaf-theoretic differential operators / 7.4:
Metric notions / 8:
Pseudo-Riemannian metrics / 8.1:
Geometry of symmetric affine connections / 8.2:
Laplacian (or isotropic) neighbours / 8.3:
The Laplace operator / 8.4:
Appendix
Category theory / A.1:
Models; sheaf semantics / A.2:
A simple topos model / A.3:
Microlinearity / A.4:
Linear algebra over local rings; Grassmannians / A.5:
Topology / A.6:
Polynomial maps / A.7:
The complex of singular cubes / A.8:
"Nullstellensatz" in multilinear algebra / A.9:
Bibliography
Index
Preface
Acknowledgements
Calculus and linear algebra / 1:
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