nLab para-complex structure

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 $M$.

Related concepts

• para-Hermitian manifold

References

General:

• Vicente Cortés, Christoph Mayer, Thomas Mohaupt, and Frank Saueressig. Special geometry of Euclidean supersymmetry 1. Vector multiplets. Journal of High Energy Physics 2004, no. 03 (2004): 028. [doi:10.1088/1126-6708/2004/03/028]