nLab para-Hermitian manifold

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)

Idea

The analogue of Hermitian manifold for para-complex structure.

Definition

Definition

An almost para-Hermitian manifold $(M,J,\eta)$ is an almost para-complex manifold $(M,J)$ endowed with a pseudo-Riemannian metric $\eta$ of signature $(d,d)$, for $\text{dim}(M)=2d$, such that

$\eta(J(X),J(Y))=-\eta(X,Y)$

References

General:

• Stefan Ivanov, Simeon Zamkovy, ParaHermitian and paraquaternionic manifolds, Differential Geometry and its Applications 23 2 (2005) 205-234 [doi:10.1016/j.difgeo.2005.06.002]

As target spacetimes in a context of type II geometry:

Last revised on November 2, 2023 at 18:51:48. See the history of this page for a list of all contributions to it.