nLab para-Hermitian manifold

Context

Complex geometry

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

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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

The analogue of Hermitian manifold for para-complex structure.

Definition

Definition

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

η(J(X),J(Y))=η(X,Y) \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.