Geometry of Mb̲ius transformations : elliptic, parabolic and hyperbolic actions of SL2(R) /

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL2(R). Starting from elementary facts in group theory, the author unveils surpr...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Kisil, Vladimir V.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Singapore ; London : World Scientific, ©2012.
Temas:
Mb̲ius transformations.
Transformations (Mathematics)
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Preface; List of Figures; 1. Erlangen Programme: Preview; 1.1 Make a Guess in Three Attempts; 1.2 Covariance of FSCc; 1.3 Invariants: Algebraic and Geometric; 1.4 Joint Invariants: Orthogonality; 1.5 Higher-order Joint Invariants: Focal Orthogonality; 1.6 Distance, Length and Perpendicularity; 1.7 The Erlangen Programme at Large; 2. Groups and Homogeneous Spaces; 2.1 Groups and Transformations; 2.2 Subgroups and Homogeneous Spaces; 2.2.1 From a Homogeneous Space to the Isotropy Subgroup; 2.2.2 From a Subgroup to the Homogeneous Space.
  • 2.3 Differentiation on Lie Groups and Lie Algebras2.3.1 One-parameter Subgroups and Lie Algebras; 2.3.2 Invariant Vector Fields and Lie Algebras; 2.3.3 Commutator in Lie Algebras; 3. Homogeneous Spaces from the Group SL2(R); 3.1 The Affine Group and the Real Line; 3.2 One-dimensional Subgroups of SL2(R); 3.3 Two-dimensional Homogeneous Spaces; 3.3.1 From the Subgroup K; 3.3.2 From the Subgroup N; 3.3.3 From the Subgroup A; 3.3.4 Unifying All Three Cases; 3.4 Elliptic, Parabolic and Hyperbolic Cases; 3.5 Orbits of the Subgroup Actions; 3.6 Unifying EPH Cases: The First Attempt.
  • 3.7 Isotropy Subgroups4. The Extended Fillmore-Springer-Cnops Construction; 4.1 Invariance of Cycles; 4.2 Projective Spaces of Cycles; 4.3 Covariance of FSCc; 4.4 Origins of FSCc; 4.4.1 Projective Coordiantes and Polynomials; 4.4.2 Co-Adjoint Representation; 4.5 Projective Cross-Ratio; 5. Indefinite Product Space of Cycles; 5.1 Cycles: An Appearance and the Essence; 5.2 Cycles as Vectors; 5.3 Invariant Cycle Product; 5.4 Zero-radius Cycles; 5.5 Cauchy-Schwarz Inequality and Tangent Cycles; 6. Joint Invariants of Cycles: Orthogonality; 6.1 Orthogonality of Cycles; 6.2 Orthogonality Miscellanea.
  • 6.3 Ghost Cycles and Orthogonality6.4 Actions of FSCc Matrices; 6.5 Inversions and Reflections in Cycles; 6.6 Higher-order Joint Invariants: Focal Orthogonality; 7. Metric Invariants in Upper Half-Planes; 7.1 Distances; 7.2 Lengths; 7.3 Conformal Properties of Mobius Maps; 7.4 Perpendicularity and Orthogonality; 7.5 Infinitesimal-radius Cycles; 7.6 Infinitesimal Conformality; 8. Global Geometry of Upper Half-Planes; 8.1 Compactification of the Point Space; 8.2 (Non)-Invariance of The Upper Half-Plane; 8.3 Optics and Mechanics; 8.3.1 Optics; 8.3.2 Classical Mechanics; 8.3.3 Quantum Mechanics.
  • 8.4 Relativity of Space-Time9. Invariant Metric and Geodesics; 9.1 Metrics, Curves' Lengths and Extrema; 9.2 Invariant Metric; 9.3 Geodesics: Additivity of Metric; 9.4 Geometric Invariants; 9.5 Invariant Metric and Cross-Ratio; 10. Conformal Unit Disk; 10.1 Elliptic Cayley Transforms; 10.2 Hyperbolic Cayley Transform; 10.3 Parabolic Cayley Transforms; 10.4 Cayley Transforms of Cycles; 10.4.1 Cayley Transform and FSSc; 10.4.2 Geodesics on the Disks; 11. Unitary Rotations; 11.1 Unitary Rotations -An Algebraic Approach; 11.2 Unitary Rotations -A Geometrical Viewpoint.

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