# nLab para-complex structure

Contents

### Context

#### Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

complex geometry

### Examples

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

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

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Intuitively, a para-complex structure is to a complex structure what the para-complex numbers? (see here for the moment) are to the complex numbers.

## Definition

### Para-complex structure on vector spaces

###### Definition

Let $V$ be a finite dimensional real vector space. A para-complex structure on $V$ is a nontrivial involution $I\in \text{End}(V)$ , i.e., $I^2=\text{Id}$ and $I\neq Id$, such that the two eigenspaces $V^{\pm}:= \text{ker}(Id\mp I)$ of $I$ are of the same dimension. A vector space $V$ endowed with a para-complex structure is known as a para-complex vector space.

###### Definition

An almost para-complex manifold is a smooth manifold $M$ with an endomorphism field $I\in \Gamma(\text{End}TM)$ such that for all $p\in M$, $I_p$ is a para-complex structure on $T_p M$. A splitting

$TM = L_{+} \oplus L_{-}$

on eigenspaces associated with eigenvalues $\pm$ of $J$ is an almost para-complex structure on $M$.

## References

General:

Last revised on November 2, 2023 at 18:43:04. See the history of this page for a list of all contributions to it.