synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
An almost Hermitian structure a reduction of the structure group along the inclusion $U(n) \hookrightarrow GL(n,\mathbb{C})$ of the unitary group into the complex general linear group.
Under further embedding $U(n) \hookrightarrow GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$ an almost hermitian structure on the frame bundle of a smooth manifold, hence a G-structure for $G = U(n)$, is first of all the choice of an almost complex structure and then an almost Hermitian manifold structure.
An first-order intgrable $U(n)$-structure (almost Hermitian manifold) structure is Kähler manifold structure.
By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that an almost Hermitian structure is precisely a joint orthogonal structure, almost symplectic structure and almost complex manifold.
Since the inclusion $U(n) \hookrightarrow GL(2n,\mathbb{R})$ factors through the symplectic group via the maximal compact subgroup inclusion
an almost Hermitian manifold structure is in particular an almost complex structure. Conversely, since the maximal compact subgroup inclusion is a homotopy equivalence, there is no obstruction to lifting an almost complex structure to an almost Hermitian structure.
An first-order integrable almost Hermitian structure is a Kähler manifold structure.