Geometry, Lie Theory and Applications: The Abel Symposium 2019

Preface
Contents
Four-Dimensional Homogeneous Generalizations of Einstein Metrics
1 Introduction
1.1 Linear Generalizations of Einstein Metrics
1.2 Weakly-Einstein Conditions
1.3 Conformal Generalizations of Einstein Metrics
1.4 Ricci Solitons
2 Homogeneous Four-Manifolds with Cyclic-Parallel Ricci Tensor
3 Weakly-Einstein Homogeneous Spaces
4 Conformally Einstein Homogeneous Spaces
5 Conformal C-Spaces
References
Conformal and Isometric Embeddings of Gravitational Instantons
1 Introduction
2 Isometric Embeddings of Class 2
2.1 Basic Theory
2.2 Necessary Conditions
2.3 Failure of Sufficiency
2.4 Obtaining Kab from X in the General Case
3 The Burns Metric
4 Conformal Isometric Embeddings of the Fubini-Study Metric
4.1 Canonical Embedding of CP2 in S7
5 Conformal Embeddings of LRS Bianchi-Type IX in R7
5.1 Isometric Embeddings in R7
5.2 LRS Bianchi-IX Isometrically Embedded into Flat R8
5.2.1 CP2
5.2.2 Eguchi-Hanson
5.2.3 Anti-Self-Dual Taub-NUT
References
Recent Results on Closed G2-Structures
1 Introduction
2 Preliminaries
3 Closed G2-Structures
4 Examples of Manifolds Admitting Closed G2-Structures
4.1 Closed G2-Structures with Symmetry
4.2 A Compact Example Obtained via the Resolution of an Orbifold
4.3 A Comparison with Lie Algebras Admitting Symplectic Structures
4.4 Products, Warped Products, and Lie Algebras Extensions
5 Some Remarks on Closed G2-Structures Induced by Coupled SU(3)-Structures on Lie Algebras
References
Almost Zoll Affine Surfaces
1 Introduction
2 The Quasi-Einstein Equation
3 The Affine Structures M(c)
4 The Geodesic Structure of M(c)
5 An Almost Zoll Structure on the Cylinder
References
Distinguished Curves and First Integrals on Poincaré–Einstein and Other Conformally Singular Geometries
1 Introduction
2 Conformal Geometry and Conformal Tractor Calculus
3 Distinguished Curves in Conformal Geometry
4 Metric Geodesics via Conformal Tractor Machinery
5 (Poincaré-)Einstein Structures
6 Conserved Quantities
6.1 An Example in the Poincaré-Einstein Case
References
A Car as Parabolic Geometry
1 Car and Engel Distribution
1.1 Configuration Space and Nonholonomic Constraints
1.2 Movement and the Role of the Tires
1.3 Velocity Distribution as an Engel Distribution
1.4 Equivalence of Engel Distributions
2 Car and Engel Distribution with a Split
2.1 Two Distinguished Directions
2.2 New Geometry: Engel Distributions with a Split
3 Explaining the so(2,3)=sp(2,R) Symmetry
3.1 A Double Fibration
3.2 Conformal Structure on myredQ
3.2.1 Geometry of Oriented Circles on the Plane
3.2.2 Conformal Minkowski Space in 3-Dimensions is SO(2,3) Symmetric
3.3 Geometry of 3rd Order ODEs
3.4 Contact Projective Geometry on darkgreenP
3.5 Chern's Double Fibration myredQM→darkgreenP, the Geometries on myredQ and darkgreenP and a Problem About a Car on a Curved Terrain
4 Lie's Correspondence
4.1 Lagrangian Planes in R4 and Oriented Circles in the Plane
4.1.1 Double Cover of SO(2,3) by Sp(2,R)
4.2 Lie's Twistor Fibration
4.3 The Picture in Terms of Parabolic Subgroups in Sp(2,R)
4.3.1 Car's Gradation in sp(2,R)
4.3.2 Parabolic Subalgebras in sp(2,R)
4.3.3 Twistor Fibration and Three Flat Parabolic Geometries Associated with a Car
4.4 Outlook: Parabolic Twistor Fibrations in Physics and in Nonholonomic Mechanics
References
Legendrian Cone Structures and Contact Prolongations
1 Introduction
2 Contact Prolongation and Contact G-Structure
3 Contact Prolongation and Legendrian Submanifolds
References
The Search for Solitons on Homogeneous Spaces
1 Introduction
2 Solitons in Differential Geometry
3 Homogeneous Geometric Structures
3.1 Tensors
3.2 Equivalence
3.3 Operators Versus Tensors
4 Solitons on Homogeneous Spaces
5 On the Evolution of Homogeneous Geometric Structures
6 The Moving-Bracket Approach
6.1 The Bracket Flow
6.2 Bracket Flow Evolution of Solitons
7 Simultaneous Diagonalization of a Homogeneous Geometric Flow
References
On Ricci Negative Lie Groups
1 Introduction
2 Some History
2.1 Negative Sectional
2.2 Negative Ricci, Non-solvable Case
2.3 Negative Ricci, Solvable Case
3 Degeneration Principle
4 Ricci Negative Derivations
4.1 Ricci Operator
4.2 Diagonalizable Ricci Negative Derivations
4.3 Strongly Ricci Negative Derivations and the Moment Map
5 An Open and Convex Cone of Ricci Negative Derivations
5.1 Nice Bases
6 Concluding Remarks
7 Appendix: The Moment Map
References
Semi-Riemannian Cones
1 Introduction
2 Preliminaries
2.1 Curvature and Geodesics of Semi-Riemannian Cones
2.2 Completeness of Certain Warped Products
3 Survey of General Results
3.1 Holonomy Groups and Gallot's Theorem
3.2 Decomposable Cones Over Complete and Over Compact Manifolds
3.3 Local Structure of Non Irreducible Cones
3.4 Holonomy of Cones
3.4.1 Irreducible Cone Holonomies
3.4.2 Holonomy of Non Irreducible, Indecomposable Cones
4 Semi-Riemannian Cones with Parallel Vector Fields
4.1 The Proof of Gallot's Theorem
4.2 A Generalisation of Gallot's Theorem
4.3 Non-Flat Cones with Parallel Vector Field
5 Lorentzian Cones and Applications to Killing Spinors
5.1 Parallel Spinors and Killing Spinors
5.2 Lorentzian Cones and Killing Spinors
References
Building New Einstein Spaces by Deforming Symmetric EinsteinSpaces
1 Introduction
2 I-Degeneracy and the Boost Weight Decomposition of Tensors
2.1 Classification of Pseudo-Riemannian Spaces
3 I-Degenerate Metrics and Their Deformations
4 Pseudo-Riemannian Spaces with Constant Scalar Curvature Invariants
5 Para-Kähler Metrics
5.1 Symmetric Metrics of Neutral Signature
5.2 I-Preserving Metric Deformations
5.3 Example: Four-Dimensional Para-Kähler Einstein Neutral Metrics
5.4 Rigidity of Higher Dimensional Para-Kähler Einstein Neutral Metrics
5.5 Almost-Para-Kähler Einstein Neutral Metrics
6 Kähler Metrics
6.1 I-Preserving Metric Deformations
6.2 Example: Kähler Einstein Metrics of Signature (2,4)
6.3 Example: Kähler Einstein Metrics of Signature (2,6)
6.4 Almost-Kähler Einstein Spaces
7 Conclusions
References
Remarks on Highly Supersymmetric Backgrounds of 11-Dimensional Supergravity
1 Introduction
2 Filtered Deformations and the Reconstruction Theorem
3 Dirac Kernels and Lie Pairs
4 An Alternative Proof of Theorem 1
5 The Reconstruction Problem for Supergravity Backgrounds
6 The SO9(R)-Orbits in 3 R9 of Subminimal Rank
7 Example
References
Krichever–Novikov Type Algebras. A General Review and the Genus Zero Case
1 Introduction
2 The Witt and the Virasoro Algebra and Their Relatives
3 The Geometric Picture
3.1 Meromorphic Forms
4 Algebraic Structures
4.1 Associative Structure
4.2 Lie and Poisson Algebra Structure
4.3 The Algebra of Differential Operators
4.4 Lie Superalgebras of Half Forms
4.5 Higher Genus Current Algebras
4.6 Krichever–Novikov Type Algebras
5 Almost-Graded Structure
5.1 Definition of Almost-Gradedness
5.2 Separating Cycle and Krichever-Novikov Pairing
5.3 The Homogeneous Subspaces
5.4 The Algebras
5.5 Triangular Decomposition and Filtrations
6 Central Extensions
6.1 Central Extensions and Cocycles
6.2 Geometric Cocycles
6.2.1 Projective and Affine Connections
6.2.2 The Function Algebra A
6.2.3 The Current Algebra g
6.2.4 The Vector Field Algebra L
6.2.5 The Differential Operator Algebra D1
6.2.6 The Lie Superalgebra S
6.3 Uniqueness and Classification of Central Extensions
7 The Genus Zero and Multi-Point Case
7.1 The Geometric Set-up
7.2 The Algebras for the Standard Splitting
7.3 Central Extensions
7.4 Universal Central Extensions
8 The Three-Point and Genus Zero Case
8.1 Symmetries
8.2 The Associative Algebra
8.3 Current and Affine Algebra
8.4 Three-Point sl(2,C)-Current Algebra for Genus 0
8.5 Vector Field Algebra
8.6 Differential Operator Algebra D1
8.7 The Lie Superalgebra
8.8 Some Remarks on the Calculations of the Residues
References