nLab para-quaternionic structure

Contents

Context

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

Manifolds and cobordisms

manifolds and cobordisms

cobordism theory, Introduction

Definitions

locally Euclidean space
manifold
- topological manifold
- differentiable manifold, ,smooth manifold
infinite dimensional manifold
- Banach manifold, Hilbert manifold, ILH manifold, Frechet manifold, convenient manifold
tangent bundle
normal bundle
G-structure, torsion of a G-structure
- orientation, spin structure, string structure, fivebrane structure
Cartan geometry:
cobordism

Genera and invariants

signature genus, Kervaire invariant
A-hat genus, Witten genus

Classification

2-manifolds/surfaces
- genus of a surface
3-manifolds
- Kirby calculus
4-manifolds
- Dehn surgery
- exotic smooth structure

Theorems

Whitney embedding theorem
Thom's transversality theorem
Pontrjagin-Thom construction
Galatius-Tillmann-Madsen-Weiss theorem
geometrization conjecture,
- Poincaré conjecture
- elliptization conjecture
cobordism hypothesis-theorem

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

differentiation, chain rule
differentiable function
infinitesimal space, infinitesimally thickened point, amazing right adjoint

V-manifolds

differentiable manifold, coordinate chart, atlas
smooth manifold, smooth structure, exotic smooth structure
analytic manifold, complex manifold
formal smooth manifold, derived smooth manifold

smooth space

diffeological space, Frölicher space
manifold structure of mapping spaces

Tangency

tangent bundle, frame bundle
- vector field, multivector field, tangent Lie algebroid;
- differential forms, de Rham complex, Dolbeault complex
local diffeomorphism, formally étale morphism
submersion, formally smooth morphism,
immersion, formally unramified morphism,
de Rham space, crystal
infinitesimal disk bundle

The magic algebraic facts

embedding of smooth manifolds into formal duals of R-algebras
smooth Serre-Swan theorem
derivations of smooth functions are vector fields

Theorems

Hadamard lemma
Borel's theorem
Boman's theorem
Whitney extension theorem
Steenrod-Wockel approximation theorem
Whitney embedding theorem
Poincare lemma
Stokes theorem
de Rham theorem
Hochschild-Kostant-Rosenberg theorem
differential cohomology hexagon

Axiomatics

Kock-Lawvere axiom

cohesion

(shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

classical modality

tangent cohesion

differential cohomology diagram

differential cohesion

(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

$(\Re \dashv \Im \dashv \&)$

graded differential cohesion

fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality

$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Models for Smooth Infinitesimal Analysis
- smooth algebra ( $C^\infty$ -ring)
- smooth locus
- Fermat theory
Cahiers topos
smooth ∞-groupoid
formal smooth ∞-groupoid
super formal smooth ∞-groupoid

Lie theory, ∞-Lie theory

Lie algebra, Lie n-algebra, L-∞ algebra
Lie group, Lie 2-group, smooth ∞-group

differential equations, variational calculus

D-geometry, D-module
jet bundle
variational bicomplex, Euler-Lagrange complex
Euler-Lagrange equation, de Donder-Weyl formalism,
phase space

Chern-Weil theory, ∞-Chern-Weil theory

connection on a bundle, connection on an ∞-bundle
differential cohomology
parallel transport, higher parallel transport, fiber integration in differential cohomology
holonomy, higher holonomy
gauge theory, higher gauge theory
Wilson line, Wilson surface

Cartan geometry (super, higher)

Klein geometry, (higher)
G-structure, torsion of a G-structure
Euclidean geometry, hyperbolic geometry, elliptic geometry
(pseudo-)Riemannian geometry
complex geometry
symplectic geometry
conformal geometry

Idea
Definition

Para-quaternionic structure on vector spaces
Para-quaternionic Kähler manifold

References

Idea

The analogue of quaternionic structure for para-quaternions?.

Definition

Para-quaternionic structure on vector spaces

Definition

A para-quaternionic structure on a vector space $V$ is a Lie subalgebra $Q\subset \text{End}(V)$ of the endomorphism Lie algebra which admits a linear basis $(J_1, J_2, J_3)$ such that $J_3 = J_1 J_2$ and $J^2_{\alpha}= \epsilon_{\alpha} Id$ , where $(\epsilon_1,\epsilon_2,\epsilon_3)=(-1,1,1)$ .

Para-quaternionic Kähler manifold

Definition

A pseudo-Riemannian manifold $(M, g)$ of dimension $\geq 5$ endowed with a parallel distribution $Q_p\subset \text{End}(T_p M)$ of $g$ -skew-symmetric para-quaternionic structures is called a para-quaternionic Kähler manifold.

The metric $g$ of a para-quaternionic Kähler manifold has signature $(2n, 2n)$ and is Einstein.

References

General:

Dmitry Vladimirovich Alekseevsky, and Vicente Cortés. The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45(1): 215-251 (March 2008).
David E. Blair, J. Davidov and O. Muskarov: Hyperbolic twistor spaces, Rocky Mountain J. Math. 35 (2005), 1437–1465.
David E. Blair. A product twistor space, Serdica Math. J. 28 (2002), 163–174.

On para-quaternionic contact structures:

Marina Tchomakova, Stefan Ivanov, Simeon Zamkovoy. Geometry of paraquaternionic contact structures (2024). (arXiv:2404.16713).

Last revised on April 26, 2024 at 08:57:50. See the history of this page for a list of all contributions to it.

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