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Global Differential Geometry / edited by Christian Bär, Joachim Lohkamp, Matthias Schwarz
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Proceedings in Mathematics ; 17 | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2012Description: Online-Ressource (VIII, 521p. 19 illus, digital)ISBN:- 9783642228421
- 516.362
- 516.3/62
- 516.36
- QA641-670
- QA670 .G56 2012
Contents:
Summary: Introduction -- Symplectic Geometry.- Geometric Analysis -- Riemann Geometry.Summary: This volume contains a collection of well-written surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.PPN: PPN: 1651230595Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Global Differential Geometry; Preface; Contents; Part I Riemannian Geometry; Holonomy Groups and Algebras; 1 Introduction; 2 Basic Definitions and Results; 2.1 Connections on Principal Bundles; 2.2 Connections on Vector Bundles; 2.3 The Spencer Complex; 2.4 G-Structures and Intrinsic Torsion; 2.5 Symmetric Connections; 2.6 Splitting Theorems; 3 Classification Results; 3.1 Irreducible Symmetric Spaces; 3.2 Holonomy of Riemannian Manifolds; 3.3 Holonomy Groups of Pseudo-Riemannian Manifolds; 3.3.1 Irreducible Holonomy Groups; 3.3.2 Indecomposable Lorentzian Holonomy Groups
3.3.3 Indecomposable Holonomy Groups of Pseudo-RiemannianManifolds of Signature (p,q) with p, q = 23.4 Special Symplectic Holonomy Groups; 3.5 Irreducible Holonomy Groups; 4 Special Symplectic Connections; References; Entropies, Volumes, and Einstein Metrics; 1 Introduction; 2 Entropies and Volumes; 2.1 Simplicial Volume; 2.2 Spherical Volume; 2.3 Volume Entropy; 2.4 Topological Entropy; 2.5 Minimal Volume; 3 Isoperimetric Constants and Minimal Eigenvalues; 3.1 Minimal Eigenvalue; 3.2 Isoperimetric Constant; 4 Einstein Metrics on Four-Manifolds; 5 Minimal Volumes and Smooth Structures
ReferencesKac-Moody Geometry; 1 What is It All About; 2 Reflection Groups; 3 The Finite Dimensional Blueprint; 3.1 Semisimple Lie Groups; 3.2 Symmetric Spaces; 3.3 Polar Representations and Isoparametric Submanifolds; 3.4 Spherical Buildings; 3.4.1 Foundations; 3.4.2 Buildings and Polar Actions; 4 Kac-Moody Groups and Their Lie Algebras; 4.1 Geometric Affine Kac-Moody Algebras; 4.2 Affine Kac-Moody Groups; 5 Kac-Moody Symmetric Spaces; 6 Polar Actions; 7 Isoparametric Submanifolds; 8 Universal Geometric Twin Buildings; 8.1 Universal BN-Pairs; 8.2 Universal Geometric Twin Buildings
8.3 Linear Representations for Universal Geometric Buildings9 Conclusion and Outlook; References; Collapsing and Almost Nonnegative Curvature; 1 Introduction; 2 Convergence and Collapsing Under a Lower Curvature Bound; 3 Constructions of Almost Nonnegative Curvature; 4 Obstructions to Almost Nonnegative Curvature; 5 Almost Nonnegative Curvature Operator; 6 Conjectures and Questions; References; Algebraic Integral Geometry; 1 Algebraic Integral Geometry; 2 Classical Integral-Geometric Formulas; 2.1 Valuations; 2.2 Intrinsic Volumes; 2.3 Kinematic Formulas; 2.4 Hadwiger's Theorem
2.5 General Hadwiger Theorem3 Algebraic Structures on Valuations; 3.1 McMullen's Decomposition; 3.2 Klain Embedding; 3.3 Schneider Embedding; 3.4 Irreducibility Theorem and Smooth Valuations; 3.5 Product; 3.6 Alesker-Poincaré Duality; 3.7 Alesker-Fourier Transform; 3.8 Convolution; 3.9 Hard Lefschetz Theorems; 4 Applications in Integral Geometry; 4.1 Abstract Hadwiger-Type Theorem; 4.2 The Kinematic Coproduct; 4.3 Fundamental Theorem of Algebraic Integral Geometry; 4.4 Additive Formulas; 5 The Hermitian Case; 5.1 ValU(n) as a Vector Space; 5.2 ValU(n) as an Algebra
5.3 Hermitian Intrinsic Volumes and Tasaki Valuations
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